cp's OEIS Frontend

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

%I A022195 #24 Jul 05 2025 05:21:17
%S A022195 1,8191,44731051,209386049731,914807651274739,3867895279362300499,
%T A022195 16094312257426532376339,66441249531569955747981459,
%U A022195 273210326382611632738979052435,1121258922081448861468067825426835,4597164868683271949171164500871212435
%N A022195 Gaussian binomial coefficients [n, 12] for q = 2.
%D A022195 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
%H A022195 Vincenzo Librandi, <a href="/A022195/b022195.txt">Table of n, a(n) for n = 12..200</a>
%F A022195 a(n) = Product_{i=1..12} (2^(n-i+1)-1)/(2^i-1), by definition. - _Vincenzo Librandi_, Aug 03 2016
%F A022195 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
%t A022195 Table[QBinomial[n, 12, 2], {n, 12, 200}] (* _Vincenzo Librandi_, Aug 03 2016 *)
%o A022195 (Sage) [gaussian_binomial(n,12,2) for n in range(12,23)] # _Zerinvary Lajos_, May 25 2009
%o A022195 (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
%o A022195 (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
%Y A022195 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).
%K A022195 nonn
%O A022195 12,2
%A A022195 _N. J. A. Sloane_
%E A022195 Offset changed by _Vincenzo Librandi_, Aug 03 2016