%I A000581 M4626 N1976
%S A000581 1,9,45,165,495,1287,3003,6435,12870,24310,43758,75582,125970,203490,
%T A000581 319770,490314,735471,1081575,1562275,2220075,3108105,4292145,5852925,
%U A000581 7888725,10518300,13884156,18156204,23535820,30260340,38608020
%N A000581 Binomial coefficients C(n,8).
%C A000581 Figurate numbers based on 8-dimensional regular simplex. - Jonathan Vos 
               Post (jvospost3(AT)gmail.com), Nov 28 2004
%C A000581 a(n) = A110555(n+1,8). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 27 2005
%C A000581 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.
%C A000581 Product of 8 consecutive numbers divided by 8! - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A000581 In this sequence there are no primes - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A000581 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
%D A000581 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000581 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000581 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 A000581 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 196.
%D A000581 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 A000581 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society 
               Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%H A000581 T. D. Noe, <a href="b000581.txt">Table of n, a(n) for n=8..1000</a>
%H A000581 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000581 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000581 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=258">
               Encyclopedia of Combinatorial Structures 258</a>
%H A000581 H. K. Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
               On Regular Polytope Numbers</a>, Journal: Proc. Amer. Math. Soc. 
               131 (2003), 65-75, as PDF file.
%H A000581 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000581 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000581 J. V. Post, <a href="http://www.magicdragon.com/poly.html">Table of Polytope 
               Numbers, Sorted, Through 1,000,000</a>.
%H A000581 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Composition.html">Link to a section of The World of Mathematics.</
               a>
%F A000581 G.f.: x^8/(1-x)^9
%F A000581 (x^8-28*x^7+322*x^6-1960*x^5+6769*x^4-13132*x^3+13068*x^2-5040*x)/40320
%F A000581 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
%p A000581 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
%p A000581 A000581:=-1/(z-1)**9; [S. Plouffe in his 1992 dissertation for offset 
               0.]
%p A000581 seq(binomial(n+8,8)*1^n,n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 23 2008
%p A000581 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]
%t A000581 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
%Y A000581 Cf. A053137, A053130.
%Y A000581 Cf. A000217, A000292, A000332, A000389, A000579, A000580.
%Y A000581 Sequence in context: A008501 A008491 A023034 this_sequence A145458 A145137 
               A144902
%Y A000581 Adjacent sequences: A000578 A000579 A000580 this_sequence A000582 A000583 
               A000584
%K A000581 nonn,easy
%O A000581 8,2
%A A000581 N. J. A. Sloane (njas(AT)research.att.com).
%E A000581 More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
%E A000581 Some formulas referring to other offsets rewritten by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jul 07 2009