A022231 Gaussian binomial coefficients [ n,2 ] for q = 7.

1, 57, 2850, 140050, 6865251, 336416907, 16484565700, 807744680100, 39579496050501, 1939395353553757, 95030372653688550, 4656488262337620150, 228167924870691555751, 11180228318776923410607, 547831187620860507371400

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..200 (first 591 terms from Todd Silvestri)
• Index entries for linear recurrences with constant coefficients, signature (57, -399, 343).

Programs

Magma

r:=2; q:=7; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 12 2016

Maple

seq((7^(n+1)-1)*(7^(n+2)-1)/288, n=0..30); # Robert Israel, Dec 16 2014

Mathematica

a[n_Integer/;n>=0]:=(7^(n+1)-1)*(7^(n+2)-1)/288 (* Todd Silvestri, Dec 16 2014 *)
Table[QBinomial[n, 2, 7], {n, 2, 20}] (* Vincenzo Librandi, Aug 12 2016 *)

PARI

Vec(1/((1-x)*(1-7*x)*(1-49*x)) + O(x^30)) \\ Michel Marcus, Dec 16 2014

PARI

r=2; q=7; for(n=r,30, print1(prod(j=1,r,(1-q^(n-j+1))/(1-q^j)), ", ")) \\ G. C. Greubel, Jun 13 2018

PARI

lista(nn, na=2, q=7) = qp=matpascal(nn+q, q); vector(nn, n, qp[n+na, n]); \\ Michel Marcus, Jun 13 2018

Sage

[gaussian_binomial(n,2,7) for n in range(2,17)] # Zerinvary Lajos, May 28 2009

Formula

G.f.: x^2/((1-x)*(1-7*x)*(1-49*x)).

a(n) = (7^(n+1)-1)*(7^(n+2)-1)/288. - Todd Silvestri, Dec 16 2014

E.g.f.: (343*exp(49*x)-56*exp(7*x)+exp(x))/288. - Robert Israel, Dec 16 2014

a(n+3) = 57*a(n+2) - 399*a(n+1) + 343*a(n). - Robert Israel, Dec 16 2014

Extensions

Offset changed by Vincenzo Librandi, Aug 12 2016