A000582 - OEIS

%I M4712 N2013 #127 Jan 06 2025 12:04:34

%S 1,10,55,220,715,2002,5005,11440,24310,48620,92378,167960,293930,

%T 497420,817190,1307504,2042975,3124550,4686825,6906900,10015005,

%U 14307150,20160075,28048800,38567100,52451256,70607460,94143280,124403620,163011640,211915132

%N a(n) = binomial coefficient C(n,9).

%C Figurate numbers based on 9-dimensional regular simplex. - _Jonathan Vos Post_, Nov 28 2004

%C Product of 9 consecutive numbers divided by 9!. - _Artur Jasinski_, Dec 02 2007

%C In this sequence there are no primes. - _Artur Jasinski_, Dec 02 2007

%C 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

%C 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

%C From _Juergen Will_, Jan 23 2016: (Start)

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

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

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

%D 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 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.

%D 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 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000582/b000582.txt">Table of n, a(n) for n = 9..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=259">Encyclopedia of Combinatorial Structures 259</a>

%H Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.

%H P. A. MacMahon, <a href="http://www.jstor.org/stable/90632?seq=1#page_scan_tab_contents">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.

%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.

%H Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Jonathan Vos Post, <a href="https://web.archive.org/web/20200219170305/http://www.magicdragon.com/poly.html">Table of Polytope Numbers, Sorted, Through 1,000,000</a>.

%H Ch. Stover and E. W. Weisstein, <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>. From MathWorld - A Wolfram Web Resource.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

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

%F a(n) = -A110555(n+1, 9). - _Reinhard Zumkeller_, Jul 27 2005

%F 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

%F Sum_{k>=9} 1/a(k) = 9/8. - _Tom Edgar_, Sep 10 2015

%F 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

%p A000582 := n->binomial(n,9): seq(A000582(n), n=9..40);

%p A000582:=1/(z-1)**10; # _Simon Plouffe_ in his 1992 dissertation (offset 0)

%p seq(binomial(n,9),n=0..29); # _Zerinvary Lajos_, Jun 23 2008, _R. J. Mathar_, Jul 07 2009

%t 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 *)

%t Table[Binomial[n, 9], {n, 9, 50}] (* _Wesley Ivan Hurt_, Jul 20 2014 *)

%o (Magma) [Binomial(n,9) : n in [9..50]]; // _Wesley Ivan Hurt_, Jul 20 2014

%o (PARI) a(n)=binomial(n,9) \\ _Charles R Greathouse IV_, Jul 21 2014

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

%K easy,nonn

%O 9,2

%A _N. J. A. Sloane_

%E Formulas referring to other offsets rewritten by _R. J. Mathar_, Jul 07 2009