A000581 a(n) = binomial coefficient C(n,8).

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, 7888725, 10518300, 13884156, 18156204, 23535820, 30260340, 38608020, 48903492, 61523748, 76904685

Offset: 8

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Figurate numbers based on 8-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004

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. - R. K. Guy, Oct 19 2007

Product of 8 consecutive numbers divided by 8!. - Artur Jasinski, Dec 02 2007

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

a(n) = number of (n-8)-digit numbers with nondescending digits. E.g., a(9) = 9 = {1,2,3,..,9}, a(10) = 45 = {11-19, 22-29, 33-39, ..., 99} [0 is counted as a zero-digit number rather than a 1-digit number]. - Toby Gottfried, Feb 14 2012

a(n) =fallfac(n, 8)/8! = binomial(n, 8) is also the number of independent components of an antisymmetric tensor of rank 8 and dimension n >= 8 (for n = 1..7 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015

Number of compositions (ordered partitions) of n+1 into exactly 9 parts. - Juergen Will, Jan 02 2016

Number of weak compositions (ordered weak partitions) of n-8 into exactly 9 parts. - Juergen Will, Jan 02 2016

Partial sums of A000580. - Art Baker, Mar 26 2019

References

• 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.
• Albert 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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• 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].
• Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 9.
• Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 258.
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
• P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
• Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
• Eric Weisstein's World of Mathematics, Composition.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000217, A000292, A000332, A000389, A000579, A000580, A053130, A053137, A254142, A001787, A242091.

Programs

Magma

[Binomial(n,8): n in [8..100]]; // Vincenzo Librandi, Apr 08 2011

Maple

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+1), n=8..40); # Zerinvary Lajos, Mar 13 2007
A000581:=-1/(z-1)**9; # Simon Plouffe in his 1992 dissertation, with offset 0
seq(binomial(n,8),n=8..40); # Zerinvary Lajos, Jun 23 2008

Mathematica

Table[Binomial[n,8],{n,8,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

PARI

a(n)=binomial(n,8) \\ Charles R Greathouse IV, Feb 14 2012

Formula

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

a(n) = A110555(n+1,8). - Reinhard Zumkeller, Jul 27 2005

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)/8!. - Artur Jasinski, Dec 02 2007

Sum_{k>=8} 1/a(k) = 8/7. - Tom Edgar, Sep 10 2015

Sum_{n>=8} (-1)^n/a(n) = A001787(8)*log(2) - A242091(8)/7! = 1024*log(2) - 74432/105 = 0.9065224171... - Amiram Eldar, Dec 10 2020

Extensions

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

Some formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009

3 more terms from William Boyles, Aug 06 2015