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Search: 1, 4, 10, 20, 35, 56
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| A000292 |
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Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
(Formerly M3382 N1363)
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+20
268
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| 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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The number of balls in a triangular pyramid in which each edge contains n+1 balls. The sum of the first n triangular numbers (A000217).
Also (1/6)*(n^3+3*n^2+2*n) is the number of ways to color vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3+2*x3+3*x1*x2)/6.
Also the convolution of the natural numbers with themselves - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
Connected with the Eulerian numbers (1,4,1) via 1*a(x-2)+4*a(x-1)+1*a(x) = x^3. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 15 2002
a(n) = sum |i-j| for all 1 <= i <= j <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2002
a(n) = sum of the all possible products p*q where (p,q) are ordered pairs and p+q = n+1. a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 29 2003
Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry (perry(AT)globalnet.co.uk), Jun 14 2003
Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 05 2004
Schlaefli symbol for this polyhedron: {3,3}
Transform of n^2 under the Riordan array (1/(1-x^2),x). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
a(n) = -A108299(n+5,6) = A108299(n+6,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
a(n) = -A110555(n+4,3). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005
a(n) is a perfect square only for n = {1, 2, 48}. a(48) = 19600 = 140^2. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 24 2006
a(n+1) is the number of terms in the expansion of (a_1+a_2+a_3+a_4)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007. (Corrected by Graeme McRae (g_m(AT)mcraefamily.com), Aug 28 2007)
This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!, over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007
a(n)=diff(S(n,x),x)|_{x=2}. First derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. W. Lang, Apr 04 2007.
If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Aug 15 2007
Number of n-permutations (n=4) of 2 objects u, v, with repetition allowed, containing exactly three (3) u's. Example: a(2)=4 because we have: uuuv, uuvu, uvuu and vuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008
Complement of A145397; A023533(a(n))=1; A014306(a(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 14 2008]
Equals row sums of triangle A152205 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
a(n) is the number of gifts received from the lyricist's true love up to and including day n in the song "The Twelve Days of Christmas". a(12)=364, almost the number of days in the year. [From Bernard Hill (bernard(AT)braeburn.co.uk), Dec 05 2008]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF2 denominators of A156925. See A157703 for background information.
(End)
Starting with "1" = row sums of triangle A158823 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]
Wiener index of the path graph P_n [From Eric W. Weisstein (eric(AT)weisstein.com), Apr 30 2009]
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)
This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative counterpart is A000178
seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50); (End)
a(n) is the number of non-decreasing, three-element permutations of n distinct numbers. [From Samuel Savitz, Sep 12 2009]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 4.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (1).
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.
D. Wells, The Penguin Dictionary of Curious and interesting Numbers, pp. 126-7 Penguin Books 1987.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients
Milan Janjic, Two Enumerative Functions
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
R. Jovanovic, First 2500 Tetrahedral numbers
Hyun Kwang Kim, On Regular Polytope Numbers
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Pyramid of 20 balls corresponding to a(3)=20.
G. Villemin's Almanach of Numbers, Nombres Tetraedriques
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).
Index entries for "core" sequences
Eric Weisstein's World of Mathematics, Wiener Index [From Eric W. Weisstein (eric(AT)weisstein.com), Apr 30 2009]
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FORMULA
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Partial sums of the triangular numbers (A000217).
G.f.: x/(1-x)^4. a(-4-n)=-a(n).
a(n)=(n+3)/n*a(n-1) - Ralf Stephan, Apr 26 2003
Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003
a(n)=C[1, 2, ]+C[2, 2]+...+C[n-1, 2]+C[n, 2]; n=5: a(5)=0+1+3+6+10=20. - Labos E. (labos(AT)ana.sote.hu), May 09 2003
a(n)=sum{k=0..n, k(n-k)} (offset 1). - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
Determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+2, i) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003
The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. Also the sum of n terms of A000217. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
a(n)=sum{k=0..floor((n-1)/2), (n-2k)^2} [offset 0]; a(n+1)=sum{k=0..n, k^2*(1-(-1)^(n+k-1))/2} [offset 0]; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
C(3+n,3)-C(2+n,2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2006
Values of the Verlinde formula for SL_2, with g=2: a(n)=sum(j=1, n-1, n/(2*sin^2(j*Pi/n))) - Simone Severini (ss54(AT)york.ac.uk), Sep 25 2006
a(n) = Sum[ Sum[ k, {k,1,m} ], {m,1,n} ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 28 2006
a(n)=Sum{k=1..n} binomial(n*k+1,n*k-1), with a(0)=0. - Paolo P. Lava (ppl(AT)spl.at), Apr 13 2007
a(n)=numbperm(n,3)/6, n>=2 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007
a(n-1) = 1/(1!*2!)*sum {1 <= x_1, x_2 <= n} |det V(x_1,x_2)| = 1/2*sum {1 <= i,j <= n} |i-j|, where V(x_1,x_2} is the Vandermonde matrix of order 2. Column 2 of A133112. - Peter Bala (pbala(AT)toucansurf.com), Sep 13 2007
Starting with "1", = binomial transform of [1, 3, 3, 1,...]; e.g. a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 04 2007
a(n) = A006503(n) - A002378(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 24 2008]
a(0)=0, a(1)=1, a(2)=4, a(3)=10, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=4. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 18 2008]
sum_{n=1..infinity} 1/a(n) = 3/2, case x=1 in Gradstein-Ryshik 1.513.7. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009]
E.g.f.:((x^3)/6+x^2+x)*exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 21 2009]
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EXAMPLE
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a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.
Consider the square array
1 2 3 4 5 6...
2 4 6 8 10 12...
3 6 9 12 16 20...
4 8 12 16 20 24...
5 10 15 20 25 30...
...
then a(n) = sum of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2003
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MAPLE
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A000292 := n->binomial(n+3, 3);
Or, f:=n->(1/6)*(n^3+3*n^2+2*n);
a:=n->sum ((j+n)*(n+2)/9, j=0..n): seq(a(n), n=0..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 17 2006
Table[((n^3 - n)/6), {n, 1, 45}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007
ZL := [S, {S=Prod(B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=3..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007
a:=n->sum(numbperm (n, 2)/6, j=0..n): seq(a(n), n=1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007
seq(numbperm (n, 3)/6, n=2..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007
seq(sum(binomial(n, k+1), k=2..2), n=2..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007
a:=n->sum(j^2-j, j=0..n): seq(a(n)/2, n=1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2008
seq(binomial(n+3, 3)*1^n, n=-1..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)
nmax:=44; for n from 0 to nmax do fz(n):=product((1-m*z)^(n+1-m), m=1..n); c(n):= abs(coeff(fz(n), z, 1)); end do: a:=n-> c(n): seq(a(n), n=0..nmax);
(End)
restart: G(x):=x^3*exp(x): f[0]:=G(x): for n from 1 to 56 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/3!, n=2..46); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]
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MATHEMATICA
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Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[50]]]]]
Table[Sum[(n - i)*i, {i, 0, n}], {n, 1, 55}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[0, 44]]]]] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
lst = {}; Do[AppendTo[lst, GegenbauerC[n, 2, 1]], {n, -1, 43}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 14 2009]
a=0; Table[(a=n^2-n+a)/2, {n, 90}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009]
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PROGRAM
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(PARI) a(n)=(n)*(n+1)*(n+2)/6
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CROSSREFS
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Sums of 2 consecutive terms give A000330.
a(3n-3)=A006566(n). A000447(n)=a(2n-2). A002492(n)=a(2n+1).
First differences give triangular numbers.
Column 0 of triangle A094415.
Cf. A000217, A001044, A003991, A061552.
Cf. A040977, A133111, A133112.
A152205 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)
Cf. A156925, A157703.
(End)
A158823 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]
Sequence in context: A138778 A038409 A090579 this_sequence A101552 A038419 A057319
Adjacent sequences: A000289 A000290 A000291 this_sequence A000293 A000294 A000295
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KEYWORD
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nonn,core,easy,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Michael Somos
Corrected PARI program. - Harry J. Smith (hjsmithh(AT)sbcglobal.net), Dec 22 2008
Multiplied g.f. with x to match the offset R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 23 2009
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| A063260 |
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Sextinomial (also called hexanomial) coefficient array. |
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+20
10
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| 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780
(list; graph; listen)
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OFFSET
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0,9
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COMMENT
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The sequence of step width of this staircase array is [1,5,5,...], hence the degree sequence for the row polynomials is [0,5,10,15,...]=A008587.
The column sequences (without leading zeros) are for k=0..5 those of the lower triangular array A007318 (Pascal) and for k=6..9: A062989, A063262-4. Row sums give A000400 (powers of 6). Central coefficients give A063419; see also A018901.
This can be used to calculate the number of occurrences of a given roll of n six-sided dice, where k is the index: k=0 being the lowest possible roll (i.e. n) and n*6 being the highest roll.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
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LINKS
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S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
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FORMULA
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G.f. for row n: (sum(x^j, j=0..5))^n.
G.f. for column k: (x^(ceiling(k/5)))*N6(k, x)/(1-x)^(k+1) with the row polynomials from the staircase array A063261(k, m).
a(n, k)=0 if n=-1 or k<0 or k >= 5*n + 1; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..5) else.
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EXAMPLE
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{1}; {1, 1, 1, 1, 1, 1}; {1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1}; ...
N6(k,x)= 1 for k=0..5; N6(6,x)= 5-10*x+10*x^2-5*x^3+x^4 (from A063261).
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CROSSREFS
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The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265.
Sequence in context: A070667 A122416 A134665 this_sequence A073793 A017891 A017881
Adjacent sequences: A063257 A063258 A063259 this_sequence A063261 A063262 A063263
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 24 2001
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EXTENSIONS
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More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002
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| A063265 |
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Septinomial (also called heptanomial) coefficient array. |
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+20
6
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| 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 116, 149, 180, 206, 224, 231, 224, 206, 180, 149, 116, 84, 56, 35
(list; graph; listen)
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OFFSET
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0,10
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COMMENT
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The sequence of step width of this staircase array is [1,6,6,...], hence the degree sequence for the row polynomials is [0,6,12,18,...]= A008588.
The column sequences (without leading zeros) are for k=0..6 those of the lower triangular array A007318 (Pascal) and for k=7..9: A063267, A063417, A063418. Row sums give A000420 (powers of 7). Central coefficients give A025012.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
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LINKS
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S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
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FORMULA
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a(n, k)=0 if n=-1 or k<0 or k >= 6*n; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..6) else.
G.f. for row n: (sum(x^j, j=0..6))^n.
G.f. for column k: (x^(ceiling(k/6)))*N7(k, x)/(1-x)^(k+1) with the row polynomials of the staircase array A063266(k, m).
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EXAMPLE
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{1}; {1, 1, 1, 1, 1, 1, 1}; {1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1}; ...
N7(k,x)= 1 for k=0..6, N7(7,x)= 6-15*x+20*x^2-15*x^3+6*x^4-x^5 (from A063266).
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CROSSREFS
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The q-nomial arrays are for q=2..6: A007318 (Pascal), A027907, A008287, A035343, A063260.
Sequence in context: A007948 A038389 A058223 this_sequence A073794 A017892 A017882
Adjacent sequences: A063262 A063263 A063264 this_sequence A063266 A063267 A063268
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 24 2001
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| A090579 |
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Number of numbers with 4 decimal digits and sum of digits = n. |
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+20
3
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| 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 279, 342, 405, 465, 519, 564, 597, 615, 615, 597, 564, 519, 465, 405, 342, 279, 219, 165, 120, 84, 56, 35, 20, 10, 4, 1
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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There are 9000 numbers with 4 decimal digits, the smallest being 1000 and the largest 9999.
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EXAMPLE
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a(2)=4: 1001, 1010, 1100, 2000.
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PROGRAM
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(PARI) b=vector(36, i, 0); for(n=1000, 9999, a=eval(Vec(Str(n))); b[sum(j=1, 4, a[j])]++); for(n=1, 36, print1(b[n], ", ")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006
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CROSSREFS
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Cf. A071817 3-digit numbers, A090580 5-digit numbers, A090581 6-digit numbers.
Sequence in context: A127764 A138778 A038409 this_sequence A000292 A101552 A038419
Adjacent sequences: A090576 A090577 A090578 this_sequence A090580 A090581 A090582
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KEYWORD
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base,fini,full,nonn
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AUTHOR
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Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 12 2004
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| A101552 |
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C(n-3,3)+C(n-7,7)+...+C(n-(4*floor((n-4)/4)+3),4*floor((n-4)/4)+3). |
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+20
2
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| 0, 0, 0, 0, 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 166, 228, 322, 484, 785, 1352, 2396, 4248, 7405, 12592, 20856, 33728, 53524, 83912, 130956, 204968, 323665, 517356, 837206, 1368108, 2248479, 3700648, 6077900, 9938488, 16164330, 26154700, 42146078
(list; graph; listen)
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| A133826 |
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Triangle whose rows are sequences of increasing and decreasing tetrahedral numbers: 1; 1,4,1; 1,4,10,4,1; ... . |
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+20
2
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| 1, 1, 4, 1, 1, 4, 10, 4, 1, 1, 4, 10, 20, 10, 4, 1, 1, 4, 10, 20, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 84, 56, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 84, 120, 84, 56, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 120, 84, 56, 35, 20
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Reading the triangle by rows produces the sequence 1,1,4,1,1,4,10,4,1,..., analogous to the Smarandache crescendo pyramidal sequence A004737.
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FORMULA
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O.g.f.: (1+qx)/((1-x)(1-qx)^3(1-q^2x)) = 1 + x(1 + 4q + q^2) + x^2(1 + 4q + 10q^2 + 4q^3 + q^4) + ... .
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EXAMPLE
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Triangle starts
1;
1, 4, 1;
1, 4, 10, 4, 1;
1, 4, 10, 20, 10, 4, 1;
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CROSSREFS
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Cf. A000292, A002415 (row sums), A004737, A124258, A133825.
Sequence in context: A080061 A124258 A001638 this_sequence A122185 A136680 A035589
Adjacent sequences: A133823 A133824 A133825 this_sequence A133827 A133828 A133829
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Peter Bala (pbala(AT)toucansurf.com), Sep 25 2007
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| A038409 |
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Coordination sequence for Zeolite Code ESV. |
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+20
1
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| 1, 4, 10, 20, 35, 56, 82, 111, 143, 180, 228, 281, 329, 382, 443, 511, 581, 651, 729, 815, 910, 1000, 1087, 1195, 1313, 1423, 1530, 1644, 1777, 1915, 2041, 2172, 2311, 2461, 2620, 2772, 2927, 3094, 3279, 3453, 3622, 3807, 3995, 4194, 4389, 4583, 4780
(list; graph; listen)
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| A138778 |
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Triangle read by rows: T(n,k)=k*binomial(n-2k,3k) (n>=5, 1<=k<=n/5). |
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+20
1
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| 1, 4, 10, 20, 35, 56, 2, 84, 14, 120, 56, 165, 168, 220, 420, 286, 924, 3, 364, 1848, 30, 455, 3432, 165, 560, 6006, 660, 680, 10010, 2145, 816, 16016, 6006, 4, 969, 24752, 15015, 52, 1140, 37128, 34320, 364
(list; graph; listen)
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OFFSET
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5,2
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COMMENT
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Row n contains floor(n/5) terms.
Row sums yield A137359.
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.
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MAPLE
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T:=proc(n, k) options operator, arrow: k*binomial(n-2*k, 3*k) end proc: for n from 5 to 22 do seq(T(n, k), k=1..(1/5)*n) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A137359.
Sequence in context: A008144 A038406 A127764 this_sequence A038409 A090579 A000292
Adjacent sequences: A138775 A138776 A138777 this_sequence A138779 A138780 A138781
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2008
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