A022190 Gaussian binomial coefficients [n, 7] for q = 2.

1, 255, 43435, 6347715, 866251507, 114429029715, 14877590196755, 1919209135381395, 246614610741341843, 31627961868755063955, 4052305562169692070035, 518946525150879134496915, 66441249531569955747981459

Offset: 7

Author

N. J. A. Sloane, Jun 14 1998

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 7..200

Crossrefs

Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).

Programs

Magma

r:=7; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Mathematica

Table[QBinomial[n, 7, 2], {n, 7, 24}] (* Vincenzo Librandi, Aug 02 2016 *)

PARI

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

Sage

[gaussian_binomial(n,7,2) for n in range(7,20)] # Zerinvary Lajos, May 25 2009

Formula

G.f.: x^7/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)*(1-32*x)*(1-64*x)*(1-128*x)). - Vincenzo Librandi, Aug 07 2016

a(n) = Product_{i=1..7} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 02 2016

G.f. with an offset of 0: exp( Sum_{n >= 1} b(8*n)/b(n)*x^n/n ) = 1 + 255*x + 43435*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Extensions

Changed offset by Vincenzo Librandi, Aug 02 2016