A337370 Expansion of sqrt(2 / ( (1-12*x+4*x^2) * (1-2*x+sqrt(1-12*x+4*x^2)) )).

1, 8, 74, 736, 7606, 80464, 864772, 9400192, 103061158, 1137528688, 12623082284, 140697113792, 1574005263676, 17663830073504, 198760191043784, 2241743315230208, 25335473017856774, 286850379192127664, 3252960763923781276, 36942512756224955456, 420084161646913792724

Offset: 0

Author

Seiichi Manyama, Aug 25 2020

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..938

Crossrefs

Column k=2 of A337369.

Cf. A337390.

Programs

Maple

Rec:= 8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0:
f:= gfun:-rectoproc({Rec,a(0)=1,a(1)=8,a(2)=74},a(n),remember):
map(f, [$0..30]); # Robert Israel, Aug 27 2020

Mathematica

a[n_] := Sum[2^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Aug 25 2020 *)

PARI

N=40; x='x+O('x^N); Vec(sqrt(2/((1-12*x+4*x^2)*(1-2*x+sqrt(1-12*x+4*x^2)))))

PARI

{a(n) = sum(k=0, n, 2^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0. - Robert Israel, Aug 27 2020

a(0) = 1, a(1) = 8 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (24*n^2-12*n-4) * a(n-1) - 4 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 31 2020