A337421 Expansion of sqrt((1-6*x+sqrt(1-4*x+36*x^2)) / (2 * (1-4*x+36*x^2))).

1, 0, -14, -48, 198, 2080, 1780, -57120, -270522, 796992, 11771676, 18981600, -314843364, -1841666112, 3400749352, 74960197312, 175979793990, -1853840247168, -13190663057780, 11783856595680, 496784970525748, 1536657455021760, -11053154849810472, -96149956882617792, 4480143410034972

Offset

0,3

Links

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

Formula

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

a(0) = 1, a(1) = 0 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (8*n^2-12*n+4) * a(n-1) - 36 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

Mathematica

a[n_] := Sum[(-2)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Aug 27 2020 *)

Prog

(PARI) N=40; x='x+O('x^N); Vec(sqrt((1-6*x+sqrt(1-4*x+36*x^2))/(2*(1-4*x+36*x^2))))
(PARI) {a(n) = sum(k=0, n, (-2)^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))}

Crossrefs

Column k=2 of A337419.

Cf. A337390.

Sequence in context: A232826 A205468 A195966 * A345724 A121202 A345691

Adjacent sequences: A337418 A337419 A337420 * A337422 A337423 A337424

Keyword

sign

Author

Seiichi Manyama, Aug 27 2020

Status

approved