A000581 - OEIS

%I M4626 N1976 #163 Feb 16 2025 08:32:21

%S 1,9,45,165,495,1287,3003,6435,12870,24310,43758,75582,125970,203490,

%T 319770,490314,735471,1081575,1562275,2220075,3108105,4292145,5852925,

%U 7888725,10518300,13884156,18156204,23535820,30260340,38608020,48903492,61523748,76904685

%N a(n) = binomial coefficient C(n,8).

%C Figurate numbers based on 8-dimensional regular simplex. - _Jonathan Vos Post_, Nov 28 2004

%C 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

%C Product of 8 consecutive numbers divided by 8!. - _Artur Jasinski_, Dec 02 2007

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

%C 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

%C 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

%C Number of compositions (ordered partitions) of n+1 into exactly 9 parts. - _Juergen Will_, Jan 02 2016

%C Number of weak compositions (ordered weak partitions) of n-8 into exactly 9 parts. - _Juergen Will_, Jan 02 2016

%C Partial sums of A000580. - _Art Baker_, Mar 26 2019

%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="/A000581/b000581.txt">Table of n, a(n) for n = 8..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 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 9.

%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.

%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=258">Encyclopedia of Combinatorial Structures 258</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 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 P. A. MacMahon, <a href="http://www.jstor.org/stable/90632">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016

%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 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Composition.html">Composition</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

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

%F a(n) = A110555(n+1,8). - _Reinhard Zumkeller_, Jul 27 2005

%F a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)/8!. - _Artur Jasinski_, Dec 02 2007

%F Sum_{k>=8} 1/a(k) = 8/7. - _Tom Edgar_, Sep 10 2015

%F 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

%p 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

%p A000581:=-1/(z-1)**9; # _Simon Plouffe_ in his 1992 dissertation, with offset 0

%p seq(binomial(n,8),n=8..40); # _Zerinvary Lajos_, Jun 23 2008

%t Table[Binomial[n,8],{n,8,50}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 22 2011 *)

%o (Magma) [Binomial(n,8): n in [8..100]]; // _Vincenzo Librandi_, Apr 08 2011

%o (PARI) a(n)=binomial(n,8) \\ _Charles R Greathouse IV_, Feb 14 2012

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

%K nonn,easy,changed

%O 8,2

%A _N. J. A. Sloane_

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

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

%E 3 more terms from _William Boyles_, Aug 06 2015