A357737 Expansion of e.g.f. sin( sqrt(3) * (exp(x) - 1) )/sqrt(3).

0, 1, 1, -2, -17, -65, -134, 331, 5797, 41092, 199621, 500731, -2996432, -58995155, -573624323, -4065029714, -19194210269, 7657775035, 1581081323122, 24363365708815, 260409006907921, 2127851409822892, 11143555796154673, -27234657667343081

Offset

0,4

Links

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

Eric Weisstein's MathWorld, Bell Polynomial.

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (-3)^(k) * Stirling2(n,2*k+1).

a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357726(k).

a(n) = ( Bell_n(sqrt(3) * i) - Bell_n(-sqrt(3) * i) )/(2 * sqrt(3) * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

Prog

(PARI) my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(3)*(exp(x)-1))/sqrt(3)))))
(PARI) a(n) = sum(k=0, (n-1)\2, (-3)^k*stirling(n, 2*k+1, 2));
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3)*I)-Bell_poly(n, -sqrt(3)*I))/(2*sqrt(3)*I));

Crossrefs

Cf. A357572, A357726.

Sequence in context: A160469 A176581 A303374 * A037420 A034721 A281708

Adjacent sequences: A357734 A357735 A357736 * A357738 A357739 A357740

Keyword

sign

Author

Seiichi Manyama, Oct 11 2022

Status

approved