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A188140 Central coefficient in (1 + x + 2^n*x^2)^n.

%I A188140 #21 Mar 09 2017 05:25:01
%S A188140 1,1,9,49,1729,31361,5613441,297047297,310070491137,43518954344449,
%T A188140 287194749188794369,100075094464681701377,4382651775014102039543809,
%U A188140 3633063629528650538489708545,1089270933915429450356021881831425
%N A188140 Central coefficient in (1 + x + 2^n*x^2)^n.
%C A188140 Variant of the central trinomial coefficients: A002426(n) = [x^n] (1+x+x^2)^n.
%H A188140 G. C. Greubel, <a href="/A188140/b188140.txt">Table of n, a(n) for n = 0..80</a>
%F A188140 a(n) = Sum_{k=0..[n/2]} C(n, k) * C(n-k, n-2*k) * 2^(n*k).
%F A188140 a(n) = [x^n] (1 + x + 2^n*x^2)^n.
%F A188140 a(n) ~ 2^((n+1)^2 / 2) / (sqrt(Pi*n)) if n is even, and a(n) ~ 2^((n^2+n-1)/2) * sqrt(n) / sqrt(Pi) if n is odd. Or a(n) ~ 2^((n+2)*(n-1)/2) * (2^(n/2+1)+n + (-1)^n * (2^(n/2+1)-n)) / sqrt(2*Pi*n). - _Vaclav Kotesovec_, Feb 12 2015
%e A188140 The coefficients of x^n in (1 + x + 2^n*x^2)^n begin:
%e A188140 n=0: [(1)];
%e A188140 n=1: [1, (1), 2];
%e A188140 n=2: [1, 2, (9), 8, 16];
%e A188140 n=3: [1, 3, 27, (49), 216, 192, 512];
%e A188140 n=4: [1, 4, 70, 196, (1729), 3136, 17920, 16384, 65536];
%e A188140 n=5: [1, 5, 170, 650, 11205, (31361), 358560, 665600, ...];
%e A188140 n=6: [1, 6, 399, 1940, 65295, 249606, (5613441), 15974784, ...];
%e A188140 n=7: [1, 7, 917, 5411, 357539, 1738261, 76854407, (297047297), ...]; ...
%e A188140 where the above central coefficients in parenthesis form this sequence.
%p A188140 a:= n-> coeff((1+x+2^n*x^2)^n, x, n):
%p A188140 seq(a(n), n=0..20);
%t A188140 Table[Sum[Binomial[n,k] * Binomial[n-k, n-2*k] * 2^(n*k), {k,0,Floor[n/2]}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 11 2015 *)
%o A188140 (PARI) {a(n)=polcoeff((1+x+2^n*x^2)^n,n)}
%o A188140 (PARI) {a(n)=sum(k=0,n\2,binomial(n, k)*binomial(n-k, n-2*k)*2^(n*k))}
%Y A188140 Cf. A188202, A002426.
%K A188140 nonn
%O A188140 0,3
%A A188140 _Paul D. Hanna_, Mar 26 2011