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A022189 Gaussian binomial coefficients [n, 6] for q = 2.

%I A022189 #24 Jul 05 2025 05:20:43
%S A022189 1,127,10795,788035,53743987,3548836819,230674393235,14877590196755,
%T A022189 955841412523283,61291693863308051,3926442969043883795,
%U A022189 251413193158549532435,16094312257426532376339,1030159771762835353435923
%N A022189 Gaussian binomial coefficients [n, 6] for q = 2.
%H A022189 Vincenzo Librandi, <a href="/A022189/b022189.txt">Table of n, a(n) for n = 6..200</a>
%F A022189 a(n) = Product_{i=1..6} (2^(n-i+1)-1)/(2^i-1), by definition. - _Vincenzo Librandi_, Aug 03 2016
%F A022189 G.f. with an offset of 0: exp( Sum_{n >= 1} b(7*n)/b(n)*x^n/n ) = 1 + 127*x + 10795*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - _Peter Bala_, Jul 01 2025
%t A022189 Table[QBinomial[n, 6, 2], {n, 6, 24}] (* _Vincenzo Librandi_, Aug 03 2016 *)
%o A022189 (Sage) [gaussian_binomial(n,6,2) for n in range(6,20)] # _Zerinvary Lajos_, May 24 2009
%o A022189 (Magma) r:=6; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 03 2016
%o A022189 (PARI) r=6; 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 A022189 Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), A006110 (k = 5), this sequence (k = 6), A022190 - A022195 (k = 7 thru 12).
%K A022189 nonn
%O A022189 6,2
%A A022189 _N. J. A. Sloane_
%E A022189 Offset changed by _Vincenzo Librandi_, Aug 03 2016