cp's OEIS Frontend

A001287 a(n) = binomial coefficient C(n,10).

%I A001287 M4794 N2046 #95 Jan 07 2025 10:52:41
%S A001287 1,11,66,286,1001,3003,8008,19448,43758,92378,184756,352716,646646,
%T A001287 1144066,1961256,3268760,5311735,8436285,13123110,20030010,30045015,
%U A001287 44352165,64512240,92561040,131128140,183579396,254186856,348330136,472733756,635745396
%N A001287 a(n) = binomial coefficient C(n,10).
%C A001287 Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].
%C A001287 Product of 10 consecutive numbers divided by 10!. - _Artur Jasinski_, Dec 02 2007
%C A001287 In this sequence only 11 is prime. - _Artur Jasinski_, Dec 02 2007
%C A001287 With a different offset, number of n-permutations (n>=10) of 2 objects: u,v, with repetition allowed, containing exactly 10 u's. Example: a(1)=11 because we have uuuuuuuuuuv, uuuuuuuuuvu, uuuuuuuuvuu, uuuuuuuvuuu, uuuuuuvuuuu, uuuuuvuuuuu, uuuuvuuuuuu, uuuvuuuuuuu, uuvuuuuuuuu, uvuuuuuuuuu and vuuuuuuuuuu. - _Zerinvary Lajos_, Aug 03 2008
%C A001287 a(9+k) is the number of times that each digit appears repeated inside a list made with all the possible base 10 numbers of k digits such that their digits are read in ascending order from left to right. - _R. J. Cano_ Jul 20 2014
%C A001287 a(n) = fallfac(n,10)/10! = binomial(n, 10) is also the number of independent components of an antisymmetric tensor of rank 10 and dimension n >= 10 (for n=1..9 this becomes 0). Here fallfac is the falling factorial. - _Wolfdieter Lang_, Dec 10 2015
%D A001287 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 A001287 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
%D A001287 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 A001287 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A001287 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001287 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001287 T. D. Noe, <a href="/A001287/b001287.txt">Table of n, a(n) for n = 10..1000</a>
%H A001287 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 A001287 Matthias Beck and Serkan Hosten, <a href="http://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.
%H A001287 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 A001287 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 A001287 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=260">Encyclopedia of Combinatorial Structures 260</a>.
%H A001287 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A001287 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 A001287 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 A001287 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A001287 a(n) = A110555(n+1,10). - _Reinhard Zumkeller_, Jul 27 2005
%F A001287 a(n+9) = n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)/10!. - _Artur Jasinski_, Dec 02 2007; _R. J. Mathar_, Jul 07 2009
%F A001287 G.f.: x^10/(1-x)^11. - _Zerinvary Lajos_, Aug 06 2008; _R. J. Mathar_, Jul 07 2009
%F A001287 Sum_{k>=10} 1/a(k) = 10/9. - _Tom Edgar_, Sep 10 2015
%F A001287 Sum_{n>=10} (-1)^n/a(n) = A001787(10)*log(2) - A242091(10)/9! = 5120*log(2) - 447047/126 = 0.9215009748... - _Amiram Eldar_, Dec 10 2020
%p A001287 seq(binomial(n,10),n=10..31); # _Zerinvary Lajos_, Aug 06 2008
%t A001287 Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)/10!, {n, 1, 100}] (* _Artur Jasinski_, Dec 02 2007 *)
%t A001287 Table[Binomial[n, 10], {n, 10, 20}] (* _Zerinvary Lajos_, Jan 31 2010 *)
%o A001287 (Magma) [Binomial(n,10): n in [10..40]]; // _Vincenzo Librandi_, Sep 11 2015
%o A001287 (PARI) a(n)=binomial(n,10) \\ _Charles R Greathouse IV_, Sep 24 2015
%o A001287 (Python)
%o A001287 A001287_list, m = [], [1]*11
%o A001287 for _ in range(10**2):
%o A001287     A001287_list.append(m[-1])
%o A001287     for i in range(10):
%o A001287         m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%Y A001287 Cf. A110555, A001787, A242091.
%K A001287 nonn,easy
%O A001287 10,2
%A A001287 _N. J. A. Sloane_
%E A001287 Formulas valid for different offsets rewritten by _R. J. Mathar_, Jul 07 2009
%E A001287 Extended by _Ray Chandler_, Oct 25 2011