A000582 a(n) = binomial coefficient C(n,9).

1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, 20160075, 28048800, 38567100, 52451256, 70607460, 94143280, 124403620, 163011640, 211915132

Offset: 9

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

Figurate numbers based on 9-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004

Product of 9 consecutive numbers divided by 9!. - Artur Jasinski, Dec 02 2007

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

a(9+n) gives the number of words with n letters over the alphabet {0,1,..,9} such that these letters are read from left to right in weakly increasing (nondecreasing) order. - R. J. Cano, Jul 20 2014

a(n) = fallfac(n, 9)/9! = binomial(n, 9) is also the number of independent components of an antisymmetric tensor of rank 9 and dimension n >= 9 (for n=1..8 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015

From Juergen Will, Jan 23 2016: (Start)

Number of compositions (ordered partitions) of n+1 into exactly 10 parts.

Number of weak compositions (ordered weak partitions) of n-9 into exactly 10 parts. (End)

Number of integers divisible by 9 in the interval [0, 10^(n-8)-1]. - Miquel Cerda, Jul 02 2017

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 = 9..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.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 259
• Milan Janjic, Two Enumerative Functions
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
• P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• 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.
• 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.
• 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.
• Ch. Stover and E. W. Weisstein, Composition. From MathWorld - A Wolfram Web Resource.
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A053138, A053131, A000581, A035927, A001787, A242091.

Programs

Magma

[Binomial(n,9) : n in [9..50]]; // Wesley Ivan Hurt, Jul 20 2014

Maple

A000582 := n->binomial(n,9): seq(A000582(n), n=9..40);
A000582:=1/(z-1)**10; # Simon Plouffe in his 1992 dissertation (offset 0)
seq(binomial(n,9),n=0..29); # Zerinvary Lajos, Jun 23 2008, R. J. Mathar, Jul 07 2009

Mathematica

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!,{n,100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 9], {n, 9, 50}] (* Wesley Ivan Hurt, Jul 20 2014 *)

PARI

a(n)=binomial(n,9) \\ Charles R Greathouse IV, Jul 21 2014

Formula

G.f.: x^9/(1-x)^10.

a(n) = -A110555(n+1, 9). - Reinhard Zumkeller, Jul 27 2005

a(n+8) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)/9!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009

Sum_{k>=9} 1/a(k) = 9/8. - Tom Edgar, Sep 10 2015

Sum_{n>=9} (-1)^(n+1)/a(n) = A001787(9)*log(2) - A242091(9)/8! = 2304*log(2) - 446907/280 = 0.9146754386... - Amiram Eldar, Dec 10 2020

Extensions

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