A337393 Expansion of sqrt((1-5*x+sqrt(1-6*x+25*x^2)) / (2 * (1-6*x+25*x^2))).

1, 1, -5, -41, -125, 131, 3301, 15625, 16115, -254525, -1813055, -4617755, 14903725, 192390589, 767919595, -28588201, -18144634861, -105011253485, -184605603311, 1406589226405, 12610893954745, 40402054036345, -63847551719825, -1340432504352485, -6346702151685475

Offset: 0

Author

Seiichi Manyama, Aug 25 2020

Keywords

• sign

Links

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

Crossrefs

Column k=1 of A337419.

Cf. A082758, A188599, A337394.

Programs

Mathematica

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

PARI

N=40; x='x+O('x^N); Vec(sqrt((1-5*x+sqrt(1-6*x+25*x^2))/(2*(1-6*x+25*x^2))))

PARI

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

Formula

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

a(0) = 1, a(1) = 1 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (12*n^2-18*n+5) * a(n-1) - 25 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020