A337421 - OEIS

%I #23 Aug 29 2020 02:18:31

%S 1,0,-14,-48,198,2080,1780,-57120,-270522,796992,11771676,18981600,

%T -314843364,-1841666112,3400749352,74960197312,175979793990,

%U -1853840247168,-13190663057780,11783856595680,496784970525748,1536657455021760,-11053154849810472,-96149956882617792,4480143410034972

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

%H Seiichi Manyama, <a href="/A337421/b337421.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,2*k).

%F a(0) = 1, a(1) = 0 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (8*n^2-12*n+4) * a(n-1) - 36 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - _Seiichi Manyama_, Aug 28 2020

%t a[n_] := Sum[(-2)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 25, 0] (* _Amiram Eldar_, Aug 27 2020 *)

%o (PARI) N=40; x='x+O('x^N); Vec(sqrt((1-6*x+sqrt(1-4*x+36*x^2))/(2*(1-4*x+36*x^2))))

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

%Y Column k=2 of A337419.

%Y Cf. A337390.

%K sign

%O 0,3

%A _Seiichi Manyama_, Aug 27 2020