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A008931 Expansion of (2/(1+sqrt(1-36*x)))^(1/3).

%I A008931 #28 Jul 30 2025 06:17:16
%S A008931 1,3,45,936,22572,592515,16434495,473825700,14058408519,426438391743,
%T A008931 13164565835421,412255067017248,13064028812911440,418149414542496168,
%U A008931 13498863325944967656,439006511643775469856,14369623854340007790108,473027210589699351461700
%N A008931 Expansion of (2/(1+sqrt(1-36*x)))^(1/3).
%H A008931 G. C. Greubel, <a href="/A008931/b008931.txt">Table of n, a(n) for n = 0..640</a>
%F A008931 From _Vladimir Reshetnikov_, Oct 12 2016: (Start)
%F A008931 a(n) = 9^n*binomial(2*n + 1/3, n)/(6*n + 1).
%F A008931 D-finite with recurrence: n*(3*n+1)*a(n) = 6*(18*n^2-21*n+5)*a(n-1). (End)
%F A008931 a(n) ~ 2^(2*n-2/3)*3^(2*n-1)/(sqrt(Pi)*n^(3/2)). - _Ilya Gutkovskiy_, Oct 13 2016
%F A008931 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^9). - _Seiichi Manyama_, Jun 20 2025
%F A008931 G.f.: 2F1(1/6, 2/3 ; 4/3 ; 36*x). - _R. J. Mathar_, Jul 30 2025
%p A008931 seq(9^n*binomial(2*n +1/3, n)/(6*n+1), n=0..20); # _G. C. Greubel_, Sep 13 2019
%t A008931 CoefficientList[Series[Surd[2/(1+Sqrt[1-36x]),3],{x,0,20}],x] (* _Harvey P. Dale_, Aug 12 2016 *)
%t A008931 Table[9^n Binomial[2 n + 1/3, n]/(6 n + 1), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 12 2016 *)
%o A008931 (PARI) my(x='x+O('x^20)); Vec((2/(1+sqrt(1-36*x)))^(1/3)) \\ _G. C. Greubel_, Apr 11 2017
%o A008931 (Magma) I:=[1]; [n le 1 select I[n] else 6*(5-21*(n-1)+18*(n-1)^2)*Self(n-1)/((n-1)*(3*n-2)): n in [1..20]]; // _G. C. Greubel_, Sep 13 2019
%o A008931 (Sage) [9^n*binomial(2*n +1/3, n)/(6*n+1) for n in (0..20)] # _G. C. Greubel_, Sep 13 2019
%o A008931 (GAP) a:=[1];; for n in [2..20] do a[n]:=6*(5-21*(n-1)+18*(n-1)^2)*a[n-1]/((n-1)*(3*n-2)); od; a; # _G. C. Greubel_, Sep 13 2019
%Y A008931 Cf. A004987, A078532, A245114, A247029, A376282, A376636.
%K A008931 nonn
%O A008931 0,2
%A A008931 _N. J. A. Sloane_