A022191 Gaussian binomial coefficients [n, 8] for q = 2.

1, 511, 174251, 50955971, 13910980083, 3675639930963, 955841412523283, 246614610741341843, 63379954960524853651, 16256896431763117598611, 4165817792093527797314451, 1066968301301093995246996371, 273210326382611632738979052435

Offset: 8

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 8..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:=8; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 03 2016

Mathematica

Table[QBinomial[n, 8, 2], {n, 8, 40}] (* Vincenzo Librandi, Aug 03 2016 *)

PARI

r=8; 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,8,2) for n in range(8,20)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 03 2016

G.f. with an offset of 0: exp( Sum_{n >= 1} b(9*n)/b(n)*x^n/n ) = 1 + 511*x +174251*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 01 2025

Extensions

Offset changed by Vincenzo Librandi, Aug 03 2016