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A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

%I A357572 #34 Feb 16 2025 08:34:04
%S A357572 0,1,1,4,19,85,406,2191,13105,84190,573121,4127521,31434184,252388957,
%T A357572 2126998693,18740283556,172134162631,1644920020417,16324076578870,
%U A357572 167938152551491,1787952325142341,19667748794844550,223217829954224029,2610546296216999197
%N A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).
%H A357572 Seiichi Manyama, <a href="/A357572/b357572.txt">Table of n, a(n) for n = 0..562</a>
%H A357572 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357572 a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * Stirling2(n,2*k+1).
%F A357572 a(n) = ( Bell_n(sqrt(3)) - Bell_n(-sqrt(3)) )/(2 * sqrt(3)), where Bell_n(x) is n-th Bell polynomial.
%F A357572 a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357615(k).
%o A357572 (PARI) a(n) = sum(k=0, (n-1)\2, 3^k*stirling(n, 2*k+1, 2));
%o A357572 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357572 a(n) = round((Bell_poly(n, sqrt(3))-Bell_poly(n, -sqrt(3)))/(2*sqrt(3)));
%Y A357572 Cf. A024429, A264037, A357598.
%Y A357572 Cf. A027710, A357615, A357737.
%K A357572 nonn
%O A357572 0,4
%A A357572 _Seiichi Manyama_, Oct 05 2022