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%I #26 Jan 25 2024 12:47:28
%S 0,1,1,5,25,117,601,3509,22457,153141,1105561,8453557,68339833,
%T 581495605,5184047961,48259748533,468040609593,4719817792565,
%U 49396003390489,535526127566773,6004124908829177,69509047405180213,829801009239621849,10202835010223731893
%N Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.
%H Seiichi Manyama, <a href="/A357598/b357598.txt">Table of n, a(n) for n = 0..558</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling2(n,2*k+1).
%F a(n) = ( Bell_n(2) - Bell_n(-2) )/4, where Bell_n(x) is n-th Bell polynomial.
%F a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A065143(k).
%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*(exp(x)-1))/2)))
%o (PARI) a(n) = sum(k=0, (n-1)\2, 4^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, 2)-Bell_poly(n, -2)))/4;
%Y Cf. A024429, A264037, A357572.
%Y Cf. A065143, A078944, A357599.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Oct 05 2022
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