7,2

Figurate numbers based on 7-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 15 of these numbers. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 28 2004

a(n) = -A110555(n+1,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

a(n) is the number of terms in the expansion of (\sum_{i=1}^8 a_i)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007

Product of seven consecutive numbers divided by 7! - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

In this sequence there are no primes - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

With a different offset, number of n-permutations (n>=7) of 2 objects: u,v with repetition allowed, containing exactly seven (7) u's. Example: a(1)=8 because we have uuuuuuuv, uuuuuuvu, uuuuuvuu, uuuuvuuu, uuuvuuuu, uuvuuuuu, uvuuuuuu and vuuuuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 23 2008

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.

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.

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=7..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].

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.

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 257

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

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

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

G.f. if offset 0: 1/(1-x)^8.

(x^7-21*x^6+175*x^5-735*x^4+1624*x^3-1764*x^2+720*x)/5040.

Convolution of the nonnegative numbers (A001477) with the sequence A000579. Also convolution of the triangular numbers (A000217) with the sequence A000332. Also convolution of the sequence {1,1,1,1,...} (A000012) with the sequence A000579. Also self-convolution of the tetrahedral numbers (A000292) - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007

a(n+4)=(1/3!)*diff(S(n,x),x$3)|_{x=2}, n>=3. One sixth of third derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. W. Lang, Apr 04 2007.

a(n)=n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)/7! - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

ZL := [S, {S=Prod(B, B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=8..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

A000580:=1/(z-1)**8; [Conjectured by S. Plouffe in his 1992 dissertation.]

seq(binomial(n+7, 7)*1^n, n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 23 2008

Table[n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)/7!, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Cf. A053136, A053129, A000579, A000581, A000582.

Cf. A000217, A000292, A000332, A000389, A000579.

Adjacent sequences: A000577 A000578 A000579 this_sequence A000581 A000582 A000583

Sequence in context: A008500 A008490 A023033 this_sequence A054470 A131123 A055910

nonn,easy

njas

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