A337394 Expansion of sqrt(2 / ( (1-6*x+25*x^2) * (1-5*x+sqrt(1-6*x+25*x^2)) )).

1, 5, 11, -29, -365, -1409, -155, 29485, 170035, 309775, -2064655, -18909175, -61552739, 81290561, 1901796395, 9145986419, 8604744275, -165227713249, -1168032362879, -2913302013175, 10702975797545, 132134872338925, 519716440255535, -109051949915065, -13098011769247075

Offset

0,2

Links

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

Formula

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

a(0) = 1, a(1) = 5 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (12*n^2-6*n-1) * a(n-1) - 25 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

Mathematica

a[n_] := Sum[(-1)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Apr 29 2021 *)

Prog

(PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1-6*x+25*x^2)*(1-5*x+sqrt(1-6*x+25*x^2)))))
(PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

Crossrefs

Column k=1 of A337464.

Cf. A188599, A273055, A337393.

Sequence in context: A234511 A053185 A358900 * A090119 A174922 A088484

Adjacent sequences: A337391 A337392 A337393 * A337395 A337396 A337397

Keyword

sign

Author

Seiichi Manyama, Aug 25 2020

Status

approved