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A377269 G.f. A(x) satisfies A(x) = (1 - 9*x*A(x))^(2/3).

%I A377269 #30 Aug 05 2025 11:18:37
%S A377269 1,-6,27,-90,189,0,-1782,6318,0,-90882,360126,0,-5985819,24931800,0,
%T A377269 -446371074,1912892355,0,-35840971530,156454458930,0,-3022929941616,
%U A377269 13367712796110,0,-264079216747476,1179032268616902,0,-23685874363658232,106533987128598645,0
%N A377269 G.f. A(x) satisfies A(x) = (1 - 9*x*A(x))^(2/3).
%H A377269 Paolo Xausa, <a href="/A377269/b377269.txt">Table of n, a(n) for n = 0..1000</a>
%H A377269 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A377269 G.f.: (1/x) * Series_Reversion( x/(1-9*x)^(2/3) ).
%F A377269 a(n) = 9^n * binomial(n/3 - 5/3,n)/(n+1).
%F A377269 From _Seiichi Manyama_, Jun 22 2025: (Start)
%F A377269 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(1/2)).
%F A377269 a(3*n+2) = 0 for n > 0. (End)
%F A377269 E.g.f.: (27*x^2 + 2*hypergeom([-2/3, 5/6], [1/3, 2/3, 2/3, 1], 4*x^3) - 12*x*hypergeom([-1/3, 7/6], [2/3, 1, 4/3, 4/3], 4*x^3))/2. - _Stefano Spezia_, Jun 22 2025
%F A377269 D-finite with recurrence n*(n-1)*a(n) -54*(2*n-1)*(n-5)*a(n-3)=0. - _R. J. Mathar_, Jul 30 2025
%t A377269 A377269[n_] := 9^n*Binomial[(n - 5)/3, n]/(n + 1);
%t A377269 Array[A377269, 35, 0] (* _Paolo Xausa_, Aug 05 2025 *)
%o A377269 (PARI) a(n) = 9^n*binomial(n/3-5/3, n)/(n+1);
%Y A377269 Cf. A376636, A377268.
%K A377269 sign,easy
%O A377269 0,2
%A A377269 _Seiichi Manyama_, Oct 22 2024