A337397 Expansion of sqrt(2 / ( (1+64*x^2) * (1-8*x+sqrt(1+64*x^2)) )).

1, 2, -34, -92, 1654, 4828, -88724, -268088, 4984486, 15361708, -287691196, -898052872, 16901635516, 53234639768, -1005474931816, -3187958034544, 60375963282182, 192405594166988, -3651655920615596, -11684176213422568, 222132094724096852, 713091439789994824, -13575872676384218776

Offset: 0

Author

Seiichi Manyama, Aug 26 2020

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=4 of A337464.

Cf. A002458, A193618, A337396.

Programs

Mathematica

a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)
CoefficientList[Series[Sqrt[2/((1+64x^2)(1-8x+Sqrt[1+64x^2]))],{x,0,30}],x] (* Harvey P. Dale, Jul 24 2021 *)

PARI

N=40; x='x+O('x^N); Vec(sqrt(2/((1+64*x^2)*(1-8*x+sqrt(1+64*x^2)))))

PARI

{a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

Formula

a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

a(0) = 1, a(1) = 2 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * 2 * a(n-1) - 64 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020