Search: id:A000332 Results 1-1 of 1 results found. %I A000332 M3853 N1578 %S A000332 0,0,0,0,1,5,15,35,70,126,210,330,495,715,1001,1365,1820,2380,3060,3876, 4845, %T A000332 5985,7315,8855,10626,12650,14950,17550,20475,23751,27405,31465,35960, %U A000332 40920,46376,52360,58905,66045,73815,82251,91390,101270,111930,123410 %N A000332 Binomial coefficients binomial(n,4). %C A000332 Number of intersection points of diagonals of convex n-gon. %C A000332 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 %C A000332 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 %C A000332 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) %C A000332 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 %C A000332 a(n) = A110555(n+1,4). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005 %C A000332 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 %C A000332 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 %C A000332 Product of four consecutive numbers divided by 24 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007 %C A000332 Only prime in this sequence is 5 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007 %C A000332 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 %C A000332 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 %C A000332 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 %C A000332 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] %C A000332 Nonzero terms = row sums of triangle A158824 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009] %C A000332 Contribution from Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009: (Start) %C A000332 Except for the 4 initial 0's, is equivalent to the sum of the tetrahedral numbers from 0 to a tetrahedral number n. %C A000332 E.g. 0 + 1 = 1, 1 + 4 = 5, 5 + 10 = 15, 15 + 20 = 35, etc. (End) %C A000332 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start) %C A000332 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 %C A000332 seq(add(add(add(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..50). (End) %D A000332 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000332 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000332 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. %D A000332 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196. %D A000332 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8. %D A000332 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. %D A000332 Norbert Kaufman and R. H. Koch, Amer. Math. Monthly, 54 (Jun, 1947), p. 344. %D A000332 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954. %D A000332 Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 53, #191 %H A000332 Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1002 %H A000332 Index entries for sequences related to linear recurrences with constant coefficients %H A000332 Milan Janjic, Two Enumerative Functions %H A000332 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]. %H A000332 X. Acloque, Polynexus Numbers. %H A000332 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000332 Th. Gruner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry %H A000332 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 254 %H A000332 Hyun Kwang Kim, On Regular Polytope Numbers %H A000332 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 %H A000332 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. %H A000332 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000332 Les Reid, Counting Triangles in an Array %H A000332 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000332 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000332 Eric Weisstein's World of Mathematics, Pentatope %F A000332 n*(n-1)*(n-2)*(n-3)/24. G.f. x^4/(1-x)^5. %F A000332 a(n)=n*a(n-1)/(n-4). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 26 2003, R. J. Mathar, Jul 07 2009 %F A000332 a(n)=sum(k=1, n-3, sum(i=1, k, i*(i+1)/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 15 2003 %F A000332 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 %F A000332 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 %F A000332 G.f.: x^4/(1-x)^5 - Jon Perry (perry(AT)globalnet.co.uk), Mar 31 2004 %F A000332 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 %F A000332 First differences of A000389(n) - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 19 2004 %F A000332 The sum of the first n tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005 %F A000332 a(n)=(n-4)(n-3)(n-2)(n-1)/24 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007 %F A000332 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 %F A000332 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] %p A000332 A000332 := n->binomial(n,4); [seq(binomial(n,4), n=0..100)]; %p A000332 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 %p A000332 with(combinat):seq(numbcomp(i+4,i), i=-3..40) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007 %p A000332 A000332:=-1/(z-1)^5; [S. Plouffe in his 1992 dissertation, sequence starting at a(4).] %p A000332 seq(sum(binomial(n,k+1),k=3..3),n=0..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007 %p A000332 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 %p A000332 seq(binomial(n+4,4)*1^n,n=0..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008 %p A000332 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] %t A000332 Table[ Binomial[n, 4], {n, 3, 45} ] %t A000332 Table[Sqrt[StirlingS2[i+1, i]*(-StirlingS1[i+3, i])/6], {i,0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007 %t A000332 Table[(n - 4)(n - 3)(n - 2)(n - 1)/24, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007 %Y A000332 Cf. A053134, A053126, A000389, A000579-A000582, A075733, A006322, A006325. %Y A000332 Cf. also A000583, A014820, A092181, A092182, A092183. %Y A000332 Partial sums of A001044. %Y A000332 Cf. A000217, A000292. %Y A000332 A158824 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009] %Y A000332 Sequence in context: A000743 A138779 A090580 this_sequence A140227 A049016 A139761 %Y A000332 Adjacent sequences: A000329 A000330 A000331 this_sequence A000333 A000334 A000335 %K A000332 nonn,easy,nice %O A000332 0,6 %A A000332 N. J. A. Sloane (njas(AT)research.att.com). %E A000332 More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000 %E A000332 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009 %E A000332 Some formulas that referred to another offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009 Search completed in 0.003 seconds