8,2

Figurate numbers based on 8-dimensional regular simplex. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 28 2004

a(n) = A110555(n+1,8). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

Comment from R. K. Guy, Oct 19, 2007: just as A005712 and A000574 are described as the coefficients of x^4 and x^5 in the expansion of (1+x+x^2)^n, so should this sequence be described as the coefficients of x^3 therein.

Product of 8 consecutive numbers divided by 8! - 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>=8) of 2 objects: u,v, with repetition allowed, containing exactly eight (8) u's. Example: a(1)=9 because we have uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu and vuuuuuuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 23 2008

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. 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=8..1000

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].

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 258

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

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.: x^8/(1-x)^9

(x^8-28*x^7+322*x^6-1960*x^5+6769*x^4-13132*x^3+13068*x^2-5040*x)/40320

a(n)=n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)/8! - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007, R. J. Mathar, Jul 07 2009

ZL := [S, {S=Prod(B, B, B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=9..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

A000581:=-1/(z-1)**9; [S. Plouffe in his 1992 dissertation for offset 0.]

seq(binomial(n+8, 8)*1^n, n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 23 2008

restart: G(x):=x^8*exp(x): f[0]:=G(x): for n from 1 to 38 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/8!, n=8..37); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)/8!, {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Dec 02 2007

Cf. A053137, A053130.

Cf. A000217, A000292, A000332, A000389, A000579, A000580.

Sequence in context: A008501 A008491 A023034 this_sequence A145458 A145137 A144902

Adjacent sequences: A000578 A000579 A000580 this_sequence A000582 A000583 A000584

nonn,easy

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

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

Some formulas referring to other offsets rewritten by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009