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A376636 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^2)^(1/3).

%I A376636 #111 Aug 04 2025 18:42:03
%S A376636 1,3,9,18,0,-162,-567,0,8019,31590,0,-520506,-2160756,0,38480265,
%T A376636 164549880,0,-3072083274,-13390246485,0,258054995016,1139882486490,0,
%U A376636 -22474826957232,-100257845970825,0,2011064804461548,9039247392729582,0,-183769714890451800
%N A376636 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^2)^(1/3).
%H A376636 Paolo Xausa, <a href="/A376636/b376636.txt">Table of n, a(n) for n = 0..1000</a>
%F A376636 a(n) = 9^n * binomial(2*n/3 + 1/3,n)/(2*n+1).
%F A376636 From _Seiichi Manyama_, Jun 20 2025: (Start)
%F A376636 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)).
%F A376636 a(3*n+1) = 0 for n > 0. (End)
%F A376636 D-finite with recurrence n*(n-2)*a(n) +54*(2*n-5)*(n-4)*a(n-3)=0. - _R. J. Mathar_, Jul 30 2025
%t A376636 A376636[n_] := 9^n*Binomial[(2*n + 1)/3, n]/(2*n + 1);
%t A376636 Array[A376636, 35, 0] (* _Paolo Xausa_, Aug 04 2025 *)
%o A376636 (PARI) a(n) = 9^n*binomial(2*n/3+1/3, n)/(2*n+1);
%Y A376636 Cf. A004987, A008931, A078532, A245114, A247029, A376282.
%Y A376636 Cf. A385117.
%K A376636 sign
%O A376636 0,2
%A A376636 _Seiichi Manyama_, Oct 23 2024