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A022192 Gaussian binomial coefficients [n, 9] for q = 2.

%I A022192 #28 Jul 05 2025 05:20:58
%S A022192 1,1023,698027,408345795,222984027123,117843461817939,
%T A022192 61291693863308051,31627961868755063955,16256896431763117598611,
%U A022192 8339787869494479328087443,4274137206973266943778085267,2189425218271613769209626653075
%N A022192 Gaussian binomial coefficients [n, 9] for q = 2.
%H A022192 Vincenzo Librandi, <a href="/A022192/b022192.txt">Table of n, a(n) for n = 9..200</a>
%F A022192 a(n) = Product_{i=1..9} (2^(n-i+1)-1)/(2^i-1), by definition. - _Vincenzo Librandi_, Aug 02 2016
%F A022192 G.f.: x^9/Product_{0<=i<=9} (1-2^i*x). - _Robert Israel_, Apr 23 2017
%F A022192 G.f. with an offset of 0: exp( Sum_{n >= 1} b(10*n)/b(n)*x^n/n ) = 1 + 1023*x + 698027*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - _Peter Bala_, Jul 01 2025
%p A022192 seq(eval(expand(QDifferenceEquations:-QBinomial(n,9,q)),q=2),n=9..50);
%t A022192 QBinomial[Range[9,20],9,2] (* _Harvey P. Dale_, Jul 24 2016 *)
%o A022192 (Sage) [gaussian_binomial(n,9,2) for n in range(9,21)] # _Zerinvary Lajos_, May 25 2009
%o A022192 (Magma) r:=9; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 03 2016
%o A022192 (PARI) r=9; 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 A022192 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).
%K A022192 nonn
%O A022192 9,2
%A A022192 _N. J. A. Sloane_
%E A022192 Offset changed by _Vincenzo Librandi_, Aug 03 2016