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%I #14 Feb 16 2025 08:34:07
%S 1,1,2,3,9,6,111,-573,7638,-95751,1450431,-24643134,468589617,
%T -9843336567,226448287794,-5662061186949,152892006728841,
%U -4434211761771978,137468475061977663,-4536657554920874181,158788359466681092966,-5875324355407515077439,229142457698060305226367
%N Expansion of e.g.f. exp( ((1+2*x)^(3/2) - 1)/3 ).
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F a(n) = Sum_{k=0..n} 2^(n-k) * Stirling1(n,k) * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * Stirling1(n,k) * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.
%F a(n) = (1/exp(1/3)) * 2^n * n! * Sum_{k>=0} binomial(3*k/2,n)/(3^k * k!).
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(((1+2*x)^(3/2)-1)/3)))
%Y Cf. A049425, A380259.
%K sign,changed
%O 0,3
%A _Seiichi Manyama_, Jan 18 2025