A337422 Expansion of sqrt((1-7*x+sqrt(1-2*x+49*x^2)) / (2 * (1-2*x+49*x^2))).

1, -1, -21, -7, 739, 1629, -26859, -118329, 922419, 6886397, -27414191, -358533429, 539620621, 17229485987, 8782716411, -769962297447, -1897237412973, 31786556599917, 149610560086113, -1182765435388341, -9268347520205991, 37049669347266471, 505738623506722431

Offset: 0

Author

Seiichi Manyama, Aug 27 2020

Keywords

• sign

Links

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

Crossrefs

Column k=3 of A337419.

Cf. A245926.

Programs

Mathematica

a[n_] := Sum[(-3)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 27 2020 *)

PARI

N=40; x='x+O('x^N); Vec(sqrt((1-7*x+sqrt(1-2*x+49*x^2))/(2*(1-2*x+49*x^2))))

PARI

{a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))}

Formula

a(n) = Sum_{k=0..n} (-3)^(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) * (4*n^2-6*n+3) * a(n-1) - 49 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020