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A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.

%I A218449 #26 Feb 16 2025 08:33:18
%S A218449 1,1,7,155,11811,3309747,3548836819,14877590196755,246614610741341843,
%T A218449 16256896431763117598611,4274137206973266943778085267,
%U A218449 4488323837657412597958687922896275,18839183877670041942218307147122500601235
%N A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.
%C A218449 Compare to: [x^n] Product_{k=0..n-1} 1+2^k*x  =  2^(n*(n-1)/2).
%H A218449 G. C. Greubel, <a href="/A218449/b218449.txt">Table of n, a(n) for n = 0..50</a>
%H A218449 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>.
%F A218449 a(n) = [x^n] Product_{k=0..n-1} 1/(1 - 2^k*x).
%F A218449 a(n) ~ c * 2^(n*(n-1)), where c = A065446. - _Vaclav Kotesovec_, Sep 22 2016
%e A218449 The coefficients in Product_{k=0..n-1} 1/(1 - 2^k*x) begin:
%e A218449 n=0: [(1)];
%e A218449 n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, ...];
%e A218449 n=2: [1, 3,(7), 15, 31, 63, 127, 255, 511, 1023, ...];
%e A218449 n=3: [1, 7, 35,(155), 651, 2667, 10795, 43435, 174251, ...];
%e A218449 n=4: [1, 15, 155, 1395,(11811), 97155, 788035, 6347715, ...];
%e A218449 n=5: [1, 31, 651, 11811, 200787,(3309747), 53743987, ...];
%e A218449 n=6: [1, 63, 2667, 97155, 3309747, 109221651,(3548836819), ...];
%e A218449 n=7: [1, 127, 10795, 788035, 53743987, 3548836819, 230674393235,(14877590196755), ...]; ...
%e A218449 the coefficients in parenthesis give the initial terms of this sequence;
%e A218449 an adjacent diagonal forms the Gaussian binomial coefficients [2*n,n] for q=2:
%e A218449 [1, 3, 35, 1395, 200787, 109221651, 230674393235, ...] = A006098.
%t A218449 Table[QBinomial[2n-1, n, 2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 12 2016 *)
%o A218449 (PARI) {a(n)=polcoeff(prod(k=0,n-1,1/(1-2^k*x +x*O(x^n))),n)}
%o A218449 for(n=0,20,print1(a(n),", "))
%Y A218449 Cf. A022166, A006098, A006125.
%K A218449 nonn
%O A218449 0,3
%A A218449 _Paul D. Hanna_, Oct 28 2012