A337466 Expansion of sqrt(2 / ( (1-4*x+36*x^2) * (1-6*x+sqrt(1-4*x+36*x^2)) )).

1, 4, -6, -120, -266, 2520, 17380, -13104, -599130, -1853544, 12391116, 108252144, 6439356, -3577917200, -14043012984, 65962248352, 730407220998, 602517029400, -22507424996420, -108316306187600, 347406564086868, 5073542740156752, 7904100039294456, -143838603813578400

Offset

0,2

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+1,2*k).

a(0) = 1, a(1) = 4 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (8*n^2-4*n) * a(n-1) - 36 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

Mathematica

a[n_] := Sum[(-2)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 24, 0] (* Amiram Eldar, Apr 29 2021 *)
CoefficientList[Series[Sqrt[2/((1-4x+36x^2)(1-6x+Sqrt[1-4x+36x^2]))], {x, 0, 40}], x] (* Harvey P. Dale, Sep 07 2023 *)

Prog

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

Crossrefs

Column k=2 of A337464.

Cf. A337370, A337421.

Sequence in context: A367878 A012934 A013165 * A052672 A137025 A355232

Adjacent sequences: A337463 A337464 A337465 * A337467 A337468 A337469

Keyword

sign

Author

Seiichi Manyama, Aug 28 2020

Status

approved