A337396 Expansion of sqrt((1-8*x+sqrt(1+64*x^2)) / (2 * (1+64*x^2))).

1, -2, -26, 76, 1222, -3772, -64676, 203992, 3607622, -11510636, -207302156, 666187432, 12142184476, -39211413464, -720760216328, 2335857124016, 43208062233158, -140406756766796, -2609918906614652, 8498967890177416, 158596941629422132, -517334728427373704, -9684521991498517112

Offset

0,2

Links

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

Formula

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

a(0) = 1, a(1) = -2 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * 2 * a(n-1) - 64 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

Mathematica

a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)

Prog

(PARI) N=40; x='x+O('x^N); Vec(sqrt((1-8*x+sqrt(1+64*x^2))/(2*(1+64*x^2))))
(PARI) {a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))}

Crossrefs

Column k=4 of A337419.

Cf. A001448, A193619, A337397.

Sequence in context: A152997 A345693 A229573 * A067204 A202077 A280212

Adjacent sequences: A337393 A337394 A337395 * A337397 A337398 A337399

Keyword

sign

Author

Seiichi Manyama, Aug 26 2020

Status

approved