A337466 - OEIS

%I #23 Sep 07 2023 16:53:11

%S 1,4,-6,-120,-266,2520,17380,-13104,-599130,-1853544,12391116,

%T 108252144,6439356,-3577917200,-14043012984,65962248352,730407220998,

%U 602517029400,-22507424996420,-108316306187600,347406564086868,5073542740156752,7904100039294456,-143838603813578400

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

%H Seiichi Manyama, <a href="/A337466/b337466.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%t 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 *)

%t 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 *)

%o (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)))))

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

%Y Column k=2 of A337464.

%Y Cf. A337370, A337421.

%K sign

%O 0,2

%A _Seiichi Manyama_, Aug 28 2020