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Expansion of e.g.f. tan(Pi/3 + x*sqrt(3)/2) / sqrt(3).
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%I #13 Jun 26 2021 14:38:20

%S 1,2,6,30,198,1638,16254,188190,2490102,37067382,613089486,

%T 11154460590,221391950598,4760331408198,110229346777374,

%U 2734768080189630,72372319913943702,2034948511063817622,60583999401612797166,1903897439808684195150,62980420349165187160998

%N Expansion of e.g.f. tan(Pi/3 + x*sqrt(3)/2) / sqrt(3).

%C If f(x) is the e.g.f. of this sequence, and if x+y+z=0, then f(x)+f(y)+f(z) = 3*f(x)*f(y)*f(z).

%H Seiichi Manyama, <a href="/A334317/b334317.txt">Table of n, a(n) for n = 0..415</a>

%F E.g.f.: tan(Pi/3 + x*sqrt(3)/2) / sqrt(3).

%F a(n+1) = (3/2) * Sum_{k=0..n} binomial(n, k) * a(k) * a(n-k), with a(0) = 1, a(1) = 2.

%F a(n) ~ 2 * n! * 3^((3*n+1)/2) / Pi^(n+1). - _Vaclav Kotesovec_, Jul 06 2020

%t a[ n_] := If[n < 0, 0, n! SeriesCoefficient[Tan[Pi/3 + Sqrt[3]/2 x]/Sqrt[3], {x, 0, n}]];

%t With[{nn=20},CoefficientList[Series[(Tan[Pi/3+x Sqrt[3]/2])/Sqrt[3],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 26 2021 *)

%o (PARI) {a(n) = my(s=quadgen(12), A); if(n < 0, 0, A = simplify(tan(s/2*x + x*O(x^n))/s); n! * polcoeff( (1 + A)/(1 - 3*A), n))};

%o (PARI) {a(n) = if(n<1, n==0, n<2, 2, n--; 3/2 * sum(k=0, n, binomial(n, k) * a(k) * a(n-k)))};

%Y Cf. A000831.

%K nonn

%O 0,2

%A _Michael Somos_, Apr 22 2020