A337393 - OEIS

%I #26 Apr 29 2021 04:34:57

%S 1,1,-5,-41,-125,131,3301,15625,16115,-254525,-1813055,-4617755,

%T 14903725,192390589,767919595,-28588201,-18144634861,-105011253485,

%U -184605603311,1406589226405,12610893954745,40402054036345,-63847551719825,-1340432504352485,-6346702151685475

%N Expansion of sqrt((1-5*x+sqrt(1-6*x+25*x^2)) / (2 * (1-6*x+25*x^2))).

%H Seiichi Manyama, <a href="/A337393/b337393.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(0) = 1, a(1) = 1 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (12*n^2-18*n+5) * a(n-1) - 25 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - _Seiichi Manyama_, Aug 28 2020

%t a[n_] := Sum[(-1)^(n-k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 25, 0] (* _Amiram Eldar_, Apr 29 2021 *)

%o (PARI) N=40; x='x+O('x^N); Vec(sqrt((1-5*x+sqrt(1-6*x+25*x^2))/(2*(1-6*x+25*x^2))))

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

%Y Column k=1 of A337419.

%Y Cf. A082758, A188599, A337394.

%K sign

%O 0,3

%A _Seiichi Manyama_, Aug 25 2020