A334317 Expansion of e.g.f. tan(Pi/3 + x*sqrt(3)/2) / sqrt(3).

1, 2, 6, 30, 198, 1638, 16254, 188190, 2490102, 37067382, 613089486, 11154460590, 221391950598, 4760331408198, 110229346777374, 2734768080189630, 72372319913943702, 2034948511063817622, 60583999401612797166, 1903897439808684195150, 62980420349165187160998

Offset

0,2

Comments

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..415

Formula

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

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

a(n) ~ 2 * n! * 3^((3*n+1)/2) / Pi^(n+1). - Vaclav Kotesovec, Jul 06 2020

Mathematica

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

Prog

(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))};
(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)))};

Crossrefs

Cf. A000831.

Sequence in context: A078700 A176719 A203000 * A104561 A211211 A127482

Adjacent sequences: A334314 A334315 A334316 * A334318 A334319 A334320

Keyword

nonn

Author

Michael Somos, Apr 22 2020

Status

approved