%I #17 Oct 14 2024 23:59:35

%S 1,0,0,0,1,1,0,0,2,4,2,0,5,15,15,5,14,56,84,56,56,210,420,420,342,834,

%T 1980,2640,2409,3795,9141,15015,16445,20449,43043,80509,104962,123838,

%U 215072,419848,630838,780572,1164228,2190552,3629704,4884100,6760390,11715210

%N G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)*A(x)^2.

%H Robert Israel, <a href="/A366589/b366589.txt">Table of n, a(n) for n = 0..4920</a>

%F G.f.: A(x) = 2 / (1+sqrt(1-4*x^4*(1+x))).

%F a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(2*k,k)/(k+1).

%F (10 + 4*n)*a(n) + (26 + 8*n)*a(n + 1) + (16 + 4*n)*a(n + 2) + (-10 - n)*a(n + 5) + (-10 - n)*a(n + 6) = 0. - _Robert Israel_, Oct 14 2024

%p f:= gfun:-rectoproc({(10 + 4*n)*a(n) + (26 + 8*n)*a(n + 1) + (16 + 4*n)*a(n + 2) + (-10 - n)*a(n + 5) + (-10 - n)*a(n + 6) = 0, a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1, a(5) = 1},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Oct 14 2024

%o (PARI) a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(2*k, k)/(k+1));

%Y Cf. A115178, A366588.

%Y Cf. A366554.

%K nonn,changed

%O 0,9

%A _Seiichi Manyama_, Oct 14 2023