A022195 Gaussian binomial coefficients [n, 12] for q = 2.

1, 8191, 44731051, 209386049731, 914807651274739, 3867895279362300499, 16094312257426532376339, 66441249531569955747981459, 273210326382611632738979052435, 1121258922081448861468067825426835, 4597164868683271949171164500871212435

Offset: 12

Author

N. J. A. Sloane

Keywords

• nonn

References

• F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 12..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 - A022194 (k = 6 thru 11).

Programs

Magma

r:=12; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Aug 03 2016

Mathematica

Table[QBinomial[n, 12, 2], {n, 12, 200}] (* Vincenzo Librandi, Aug 03 2016 *)

PARI

r=12; 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,12,2) for n in range(12,23)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..12} (2^(n-i+1)-1)/(2^i-1), by definition. - Vincenzo Librandi, Aug 03 2016

G.f. assuming an offset of 0: exp( Sum_{n >= 1} b(13*n)/b(n)*x^n/n ) = 1 + 8191*x + 44731051*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - Peter Bala, Jul 03 2025

Extensions

Offset changed by Vincenzo Librandi, Aug 03 2016