A357615 Expansion of e.g.f. cosh(sqrt(3) * (exp(x) - 1)).

1, 0, 3, 9, 30, 135, 705, 3906, 22953, 145053, 985800, 7136613, 54544485, 437961888, 3685605735, 32441696325, 297977767662, 2848636972971, 28278241848309, 290931124989546, 3097051613077269, 34064462020306473, 386600759467746528, 4521440483724439521

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..562

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^k * Stirling2(n,2*k).

a(n) = ( Bell_n(sqrt(3)) + Bell_n(-sqrt(3)) )/2, where Bell_n(x) is n-th Bell polynomial.

a(n) = 1; a(n) = 3 * Sum_{k=0..n-1} binomial(n-1, k) * A357572(k).

Example

G.f. = 1 + 3*x^2 + 9*x^3 + 30*x^4 + 135*x^5 + 705*x^6 + ... - Michael Somos, Oct 06 2022

Mathematica

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Cosh[Sqrt[3] * (Exp@x - 1)], {x, 0, n}]]; (* Michael Somos, Oct 06 2022 *)

Prog

(PARI) a(n) = sum(k=0, n\2, 3^k*stirling(n, 2*k, 2));
(PARI) my(x='x+O('x^30)); apply(round, Vec(serlaplace(cosh(sqrt(3) * (exp(x) - 1))))) \\ Michel Marcus, Oct 06 2022
(PARI)  {a(n) = if(n<0, 0, n!*simplify(polcoeff( cosh(quadgen(12) * (exp(x + x*O(x^n)) - 1)), n)))}; /* Michael Somos, Oct 06 2022 */

Crossrefs

Column k=3 of A357681.

Cf. A065143, A264036.

Cf. A357572.

Sequence in context: A297202 A370439 A003604 * A292300 A058148 A031031

Adjacent sequences: A357612 A357613 A357614 * A357616 A357617 A357618

Keyword

nonn

Author

Seiichi Manyama, Oct 06 2022

Status

approved