0,6

Number of intersection points of diagonals of convex n-gon.

Also the number of equilateral triangles with vertices in a equilateral triangular array of points with n rows (offset 1), with any orientation. - Ignacio Larrosa Canestro (ignacio.larrosa(AT)eresmas.net), Apr 09 2002

Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - rgwv, Aug 02 2002

For n>0 a(n)=(-1/8)*coefficient of x in Zagier's polynomial P_(2n,n) (Zagier's polynomials are used by pari-gp for acceleration of alternating or positive series)

Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n-1)*(n-2)*(n-3))/4!) - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, R. J. Mathar, Jul 07 2009

a(n) = A110555(n+1,4). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

Maximal number of crossings that can be created by connecting n vertices with straight lines. - Cameron Redsell-Montgomerie (credsell(AT)uoguelph.ca), Jan 30 2007

If X is an n-set and Y a fixed (n-1)-subset of X then a(n) is equal to the number of 4-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Aug 15 2007

Product of four consecutive numbers divided by 24 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Only prime in this sequence is 5 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

For strings consisting entirely of 0s and 1s, the number of unique arrangements of four 1s such that 1s are not adjacent. The shortest possible string is 7 characters, of which there is only one solution: 1010101, corresponding to a(5). An eight character string has 5 solutions, nine has 15, ten has 35 and so on, congruent to A000332. - Gil Broussard (gil_broussard(AT)bellsouth.net), Mar 19 2008

Apart from the first 4 zeros, this sequence also represents the partial sums of oblong numbers. That is, a(n)=n(n-1)(n-2)(n-3)/24. - Mohammad K. Azarian (azarian(AT)evansville.edu), Apr 03 2008, R. J. Mathar, Jul 07 2009

With a different offset, number of n-permutations (n=5) of 2 objects: u,v, with repetition allowed, containing exactly four (4) u's. Example: a(5)=5 because we have uuuuv uuuvu uuvuu uvuuu vuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008

For a(n)>0, a(n) is pentagonal if and only if 3 does not divide n. All terms belong to the generalized pentagonal sequence (A001318). Cf. A000326, A145919, A145920. [From Matthew Vandermast (ghodges14(AT)comcast.net), Oct 28 2008]

Nonzero terms = row sums of triangle A158824 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]

Contribution from Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009: (Start)

Except for the 4 initial 0's, is equivalent to the sum of the tetrahedral numbers from 0 to a tetrahedral number n.

E.g. 0 + 1 = 1, 1 + 4 = 5, 5 + 10 = 15, 15 + 20 = 35, etc. (End)

Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)

If the first 3 zeros are disregarded, that is if one looks at binomial(n+3, 4) with n>=0, then it becomes a 'Matryoshka doll' sequence with alpha=0

seq(add(add(add(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..50). (End)

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. 196.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.

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. 7.

Norbert Kaufman and R. H. Koch, Amer. Math. Monthly, 54 (Jun, 1947), p. 344.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 53, #191

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1002

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].

X. Acloque, Polynexus Numbers.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Th. Gruner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 254

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

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Les Reid, Counting Triangles in an Array

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Pentatope

n*(n-1)*(n-2)*(n-3)/24. G.f. x^4/(1-x)^5.

a(n)=n*a(n-1)/(n-4). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 26 2003, R. J. Mathar, Jul 07 2009

a(n)=sum(k=1, n-3, sum(i=1, k, i*(i+1)/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 15 2003

Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...} - Jon Perry (perry(AT)globalnet.co.uk), Jun 25 2003

a(n+1) = [(n^5-(n-1)^5)-(n^3-(n-1)^3)]/24 - (n^5-(n-1)^5-1)/30; a(n) = A006322 - A006325. - Xavier Acloque Oct 20 2003, R. J. Mathar, Jul 07 2009

G.f.: x^4/(1-x)^5 - Jon Perry (perry(AT)globalnet.co.uk), Mar 31 2004

a(4n+2) = Pyr(n+4, 4n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = [(A-2)*B^3 + 3*B^2 - (A-5)*B]/6; For all positive integers i and the pentagonal number function P(x) = x*(3*x-1)/2: a(3i-2) = P(P(i)) and a(3i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 15 2004

First differences of A000389(n) - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 19 2004

The sum of the first n tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005

a(n)=(n-4)(n-3)(n-2)(n-1)/24 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Starting (1, 5, 15, 35,...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2007

sum_{n=4..infinity} 1/a(n) = 4/3, from the Taylor expansion of (1-x)^3*log(1-x) in the limit x->1. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009]

A000332 := n->binomial(n, 4); [seq(binomial(n, 4), n=0..100)];

ZL := [S, {S=Prod(B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=1..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

with(combinat):seq(numbcomp(i+4, i), i=-3..40) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007

A000332:=-1/(z-1)^5; [S. Plouffe in his 1992 dissertation, sequence starting at a(4).]

seq(sum(binomial(n, k+1), k=3..3), n=0..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007

a:=n->sum(j^3-j, j=0..n): seq(a(n)/6, n=-2..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2008

seq(binomial(n+4, 4)*1^n, n=0..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008

restart: G(x):=x^4*exp(x): f[0]:=G(x): for n from 1 to 43 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/4!, n=0..43); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

Table[ Binomial[n, 4], {n, 3, 45} ]

Table[Sqrt[StirlingS2[i+1, i]*(-StirlingS1[i+3, i])/6], {i, 0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

Table[(n - 4)(n - 3)(n - 2)(n - 1)/24, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Cf. A053134, A053126, A000389, A000579-A000582, A075733, A006322, A006325.

Cf. also A000583, A014820, A092181, A092182, A092183.

Partial sums of A001044.

Cf. A000217, A000292.

A158824 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]

Sequence in context: A000743 A138779 A090580 this_sequence A140227 A049016 A139761

Adjacent sequences: A000329 A000330 A000331 this_sequence A000333 A000334 A000335

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009

Some formulas that referred to another offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009