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

%I A357737 #16 Feb 16 2025 08:34:04
%S A357737 0,1,1,-2,-17,-65,-134,331,5797,41092,199621,500731,-2996432,
%T A357737 -58995155,-573624323,-4065029714,-19194210269,7657775035,
%U A357737 1581081323122,24363365708815,260409006907921,2127851409822892,11143555796154673,-27234657667343081
%N A357737 Expansion of e.g.f. sin( sqrt(3) * (exp(x) - 1) )/sqrt(3).
%H A357737 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357737 a(n) = Sum_{k=0..floor((n-1)/2)} (-3)^(k) * Stirling2(n,2*k+1).
%F A357737 a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357726(k).
%F A357737 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 A357737 (PARI) my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(3)*(exp(x)-1))/sqrt(3)))))
%o A357737 (PARI) a(n) = sum(k=0, (n-1)\2, (-3)^k*stirling(n, 2*k+1, 2));
%o A357737 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357737 a(n) = round((Bell_poly(n, sqrt(3)*I)-Bell_poly(n, -sqrt(3)*I))/(2*sqrt(3)*I));
%Y A357737 Cf. A357572, A357726.
%K A357737 sign
%O A357737 0,4
%A A357737 _Seiichi Manyama_, Oct 11 2022