5,2

Number of inequivalent ways of coloring the vertices of a regular 4-dimensional simplex with n colors, under the full symmetric group S_5 of order 120, with cycle index (x1^5 + 10*x1^3*x2 + 20*x1^2*x3 + 15*x1*x2^2 + 30*x1*x4 + 20*x2*x3 + 24*x5)/120.

Figurate numbers based on 5-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 10 of these 5-simplex(n) numbers (compared with g=3 for triangular numbers, g=5 for tetrahedral numbers and g=8 for pentatope numbers). - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 28 2004

a(n) = -A110555(n+1,5). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 27 2005

The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which themselves are the convolution of the nonnegative integers with themselves (making appropriate allowances for offsets of all sequences). - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006

a(n) is the number of terms in the expansion of (a_1+a_2+a_3+a_4+a_5+a_6)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007

a(n)=A052787(n+5)/120. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

Product of five consecutive numbers divided by 120 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

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. 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.

Gupta, Hansraj; Partitions of $j$-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).

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

T. D. Noe, Table of n, a(n) for n=5..1000

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, December 1972 [alternative scanned copy].

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 255

H. K. Kim, On Regular Polytope Numbers, Jounal: Proc. Amer.Math. Soc. 131 (2003), 65-75, as PDF file.

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.

J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

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

G.f. if offset 0: 1/(1-x)^6.

a(n) = (x^5-10*x^4+35*x^3-50*x^2+24*x)/120.

a(n) = (1/120)*(n^5+10*n^4+35*n^3+50*n^2+24*n). (Replace all x_i's in the cycle index by n.)

a(n+2) = sum_{i+j+k=n} ijk. - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 01 2002

Convolution of triangular numbers (A000217) with themselves

Partial sums of A000332(n) - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 19 2004

a(n)=n(n+1)(n+2)(n+3)(n+4)/120

a(n+3)=(1/2!)*diff(S(n,x),x$2)|_{x=2}, n>=2, One half of second derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. W. Lang, Apr 04 2007.

a(n)=numbperm (n,5)/120, n>=5 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

a(n)=n(n+1)(n+2)(n+3)(n+4)/120 - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

f:=n->(1/120)*(n^5+10*n^4+35*n^3+50*n^2+24*n);

ZL := [S, {S=Prod(B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=6..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

seq(numbperm (n, 5)/120, n=5..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

with(combinat):seq(numbcomp(i+5, i), i=1..37) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007

A000389:=1/(z-1)**6; [S. Plouffe, 1992 dissertation.]

seq(sum(binomial(n, k+1), k=4..4), n=5..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007

Table[Binomial[n, 5], {n, 5, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 02 2006

Table[n(n + 1)(n + 2)(n + 3)(n + 4)/120, {n, 1, 30}] - Artur Jasinski (grafix(AT)csl.pl), Nov 14 2006

Table[n(n + 1)(n + 2)(n + 3)(n + 4)/120, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

(PARI) conv(u, v)=local(w); w=vector(length(u), i, sum(j=1, i, u[j]*v[i+1-j])); w; t(n)=n*(n+1)/2; u=vector(10, i, t(i)); conv(u, u)

Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.

Cf. A000217, A005583, A051747, A000292, A000332.

Adjacent sequences: A000386 A000387 A000388 this_sequence A000390 A000391 A000392

Sequence in context: A120478 A023031 A090581 this_sequence A006090 A100356 A137361

nonn,easy,nice

njas

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000