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A001287 a(n) = binomial coefficient C(n,10).
(Formerly M4794 N2046)
20

%I M4794 N2046 #90 Dec 15 2023 15:57:35

%S 1,11,66,286,1001,3003,8008,19448,43758,92378,184756,352716,646646,

%T 1144066,1961256,3268760,5311735,8436285,13123110,20030010,30045015,

%U 44352165,64512240,92561040,131128140,183579396,254186856,348330136,472733756,635745396

%N a(n) = binomial coefficient C(n,10).

%C Coordination sequence for 10-dimensional cyclotomic lattice Z[zeta_11].

%C Product of 10 consecutive numbers divided by 10!. - _Artur Jasinski_, Dec 02 2007

%C In this sequence only 11 is prime. - _Artur Jasinski_, Dec 02 2007

%C 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 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 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 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="/A001287/b001287.txt">Table of n, a(n) for n = 10..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 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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=260">Encyclopedia of Combinatorial Structures 260</a>.

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.

%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 <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 a(n) = A110555(n+1,10). - _Reinhard Zumkeller_, Jul 27 2005

%F 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 G.f.: x^10/(1-x)^11. - _Zerinvary Lajos_, Aug 06 2008; _R. J. Mathar_, Jul 07 2009

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

%F 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 seq(binomial(n,10),n=10..31); # _Zerinvary Lajos_, Aug 06 2008

%t 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 Table[Binomial[n, 10], {n, 10, 20}] (* _Zerinvary Lajos_, Jan 31 2010 *)

%o (Magma) [Binomial(n,10): n in [10..40]]; // _Vincenzo Librandi_, Sep 11 2015

%o (PARI) a(n)=binomial(n,10) \\ _Charles R Greathouse IV_, Sep 24 2015

%o (Python)

%o A001287_list, m = [], [1]*11

%o for _ in range(10**2):

%o A001287_list.append(m[-1])

%o for i in range(10):

%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016

%Y Cf. A110555, A001787, A242091.

%K nonn,easy

%O 10,2

%A _N. J. A. Sloane_

%E Formulas valid for different offsets rewritten by _R. J. Mathar_, Jul 07 2009

%E Extended by _Ray Chandler_, Oct 25 2011

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)