cp's OEIS Frontend

A366589 G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)*A(x)^2.

%I A366589 #17 Oct 14 2024 23:59:35
%S A366589 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 A366589 1980,2640,2409,3795,9141,15015,16445,20449,43043,80509,104962,123838,
%U A366589 215072,419848,630838,780572,1164228,2190552,3629704,4884100,6760390,11715210
%N A366589 G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)*A(x)^2.
%H A366589 Robert Israel, <a href="/A366589/b366589.txt">Table of n, a(n) for n = 0..4920</a>
%F A366589 G.f.: A(x) = 2 / (1+sqrt(1-4*x^4*(1+x))).
%F A366589 a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(2*k,k)/(k+1).
%F A366589 (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 A366589 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 A366589 map(f, [$0..30]); # _Robert Israel_, Oct 14 2024
%o A366589 (PARI) a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(2*k, k)/(k+1));
%Y A366589 Cf. A115178, A366588.
%Y A366589 Cf. A366554.
%K A366589 nonn
%O A366589 0,9
%A A366589 _Seiichi Manyama_, Oct 14 2023