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

1, 1, 9, 49, 1729, 31361, 5613441, 297047297, 310070491137, 43518954344449, 287194749188794369, 100075094464681701377, 4382651775014102039543809, 3633063629528650538489708545, 1089270933915429450356021881831425

Offset: 0

Author

Paul D. Hanna, Mar 26 2011

Keywords

• nonn

Comments

Variant of the central trinomial coefficients: A002426(n) = [x^n] (1+x+x^2)^n.

Examples

			The coefficients of x^n in (1 + x + 2^n*x^2)^n begin:
n=0: [(1)];
n=1: [1, (1), 2];
n=2: [1, 2, (9), 8, 16];
n=3: [1, 3, 27, (49), 216, 192, 512];
n=4: [1, 4, 70, 196, (1729), 3136, 17920, 16384, 65536];
n=5: [1, 5, 170, 650, 11205, (31361), 358560, 665600, ...];
n=6: [1, 6, 399, 1940, 65295, 249606, (5613441), 15974784, ...];
n=7: [1, 7, 917, 5411, 357539, 1738261, 76854407, (297047297), ...]; ...
where the above central coefficients in parenthesis form this sequence.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..80

Crossrefs

Cf. A188202, A002426.

Programs

Maple

a:= n-> coeff((1+x+2^n*x^2)^n, x, n):
seq(a(n), n=0..20);

Mathematica

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 *)

PARI

{a(n)=polcoeff((1+x+2^n*x^2)^n,n)}

PARI

{a(n)=sum(k=0,n\2,binomial(n, k)*binomial(n-k, n-2*k)*2^(n*k))}

Formula

a(n) = Sum_{k=0..[n/2]} C(n, k) * C(n-k, n-2*k) * 2^(n*k).

a(n) = [x^n] (1 + x + 2^n*x^2)^n.

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