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%I #20 Apr 29 2021 04:35:08
%S 1,5,11,-29,-365,-1409,-155,29485,170035,309775,-2064655,-18909175,
%T -61552739,81290561,1901796395,9145986419,8604744275,-165227713249,
%U -1168032362879,-2913302013175,10702975797545,132134872338925,519716440255535,-109051949915065,-13098011769247075
%N Expansion of sqrt(2 / ( (1-6*x+25*x^2) * (1-5*x+sqrt(1-6*x+25*x^2)) )).
%H Seiichi Manyama, <a href="/A337394/b337394.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+1,2*k).
%F a(0) = 1, a(1) = 5 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (12*n^2-6*n-1) * a(n-1) - 25 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - _Seiichi Manyama_, Aug 29 2020
%t a[n_] := Sum[(-1)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 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(2/((1-6*x+25*x^2)*(1-5*x+sqrt(1-6*x+25*x^2)))))
%o (PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}
%Y Column k=1 of A337464.
%Y Cf. A188599, A273055, A337393.
%K sign
%O 0,2
%A _Seiichi Manyama_, Aug 25 2020
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