A357737 - OEIS

%I #15 Oct 12 2022 11:27:40

%S 0,1,1,-2,-17,-65,-134,331,5797,41092,199621,500731,-2996432,

%T -58995155,-573624323,-4065029714,-19194210269,7657775035,

%U 1581081323122,24363365708815,260409006907921,2127851409822892,11143555796154673,-27234657667343081

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

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

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

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

%F 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.

%o (PARI) my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(3)*(exp(x)-1))/sqrt(3)))))

%o (PARI) a(n) = sum(k=0, (n-1)\2, (-3)^k*stirling(n, 2*k+1, 2));

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

%o a(n) = round((Bell_poly(n, sqrt(3)*I)-Bell_poly(n, -sqrt(3)*I))/(2*sqrt(3)*I));

%Y Cf. A357572, A357726.

%K sign

%O 0,4

%A _Seiichi Manyama_, Oct 11 2022