A010971 a(n) = binomial(n,18).

1, 19, 190, 1330, 7315, 33649, 134596, 480700, 1562275, 4686825, 13123110, 34597290, 86493225, 206253075, 471435600, 1037158320, 2203961430, 4537567650, 9075135300, 17672631900, 33578000610, 62359143990, 113380261800, 202112640600, 353697121050, 608359048206

Offset: 18

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Coordination sequence for 18-dimensional cyclotomic lattice Z[zeta_19].

Product of 18 consecutive numbers divided by 18!. - Artur Jasinski, Dec 02 2007

In this sequence only 19 is prime. - Artur Jasinski, Dec 02 2007

With a different offset, number of n-permutations (n>=18) of 2 objects: u,v, with repetition allowed, containing exactly (18) u's. - Zerinvary Lajos, Aug 04 2008

Links

• T. D. Noe, Table of n, a(n) for n = 18..1000
• Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
• Milan Janjic, Two Enumerative Functions.
• Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Crossrefs

Cf. A001787, A242091.

Programs

Magma

[Binomial(n, 18): n in [18..50]]; // Vincenzo Librandi, Aug 08 2017

Maple

seq(binomial(n,18),n=18..38); # Zerinvary Lajos, Aug 04 2008

Mathematica

Table[n(n+1)(n+2)(n+3)(n+4)(n+5)(n+6)(n+7)(n+8)(n+9)(n+10)(n+11)(n+12)(n+13)(n+14)(n+15)(n+16)(n+17)/18!,{n,1,100}] (* Artur Jasinski, Dec 02 2007 *)
Table[Binomial[n, 18], {n, 18, 50}] (* Vincenzo Librandi, Aug 08 2017 *)

PARI

for(n=18,50, print1(binomial(n,18), ", ")) \\ G. C. Greubel, Nov 23 2017

Formula

a(n+17) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11)*(n+12)*(n+13)*(n+14)*(n+15)*(n+16)*(n+17)/18!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009

G.f.: x^18/(1-x)^19. - Zerinvary Lajos, Aug 04 2008; R. J. Mathar, Jul 07 2009

From Amiram Eldar, Dec 10 2020: (Start)

Sum_{n>=18} 1/a(n) = 18/17.

Sum_{n>=18} (-1)^n/a(n) = A001787(18)*log(2) - A242091(18)/17! = 2359296*log(2) - 556571077357/340340 = 0.9519925176... (End)

Extensions

Some formulas adjusted to the offset by R. J. Mathar, Jul 07 2009