A357537 a(n) = 2*A080635(n) if n > 0. a(0) = 1.

1, 2, 2, 6, 18, 78, 378, 2214, 14562, 108702, 897642, 8171766, 81066258, 871695918, 10091490138, 125189658054, 1656458307522, 23288226400062, 346663764078282, 5447099463010326, 90094171024954098, 1564653992673809358, 28467075416816935098, 541467979789775621094

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

E.g.f.: sqrt(3) tan(Pi/6 + x sqrt(3)/2).

E.g.f. A(x) satisfies 2*A' = 3 + A^2, A'' = A*A'.

Let f(x) = 1 + x + x^2. Then a(n+1) = (f(x)*d/dx)^n f'(x) evaluated at x = 0.

Example

G.f. = 1 + 2*x + 2*x^2 + 6*x^3 + 18*x^4 + 78*x^5 + 378*x^6 + ...
E.g.f. = 1 + 2*x + x^2 + x^3 + 3/4*x^4 + 13/20*x^5 + 21/40*x^6 + ...

Mathematica

a[ n_] := If[ n < 0, 0, n! Simplify@SeriesCoefficient[ Sqrt[3] Tan[ Pi/6 + x Sqrt[3]/2], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, Nest[Expand[(1 + x + x^2) D[#, x]]&, 1 + 2 x, n] /. x->0];

Prog

(PARI) {a(n) = my(A); if(n<0, 0, A = 1 + 2*x; for( k=1, n, A = A' * (1 + x + x^2)); polcoeff(A, 0))};

Crossrefs

Cf. A080635.

Sequence in context: A253284 A176754 A365103 * A173098 A370836 A233113

Adjacent sequences: A357534 A357535 A357536 * A357538 A357539 A357540

Keyword

nonn

Author

Michael Somos, Oct 02 2022

Status

approved