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%I #14 Oct 15 2022 08:08:44
%S 1,0,0,2,12,70,454,3332,27552,254400,2598852,29125932,355455468,
%T 4693396656,66671326176,1013916648840,16436063079552,282920894841096,
%U 5153797995148296,99052313167391760,2003040751641857856,42513854724369719136,944959706480298199824
%N a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(-2^(1/3) * log(1-x)).
%F a(n) = ( (2^(1/3))_n + (2^(1/3)*w)_n + (2^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.
%o (PARI) a(n) = sum(k=0, n\3, 2^k*abs(stirling(n, 3*k, 1)));
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k)/(3*k)!)))
%o (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round(Pochhammer(v, n)+Pochhammer(v*w, n)+Pochhammer(v*w^2, n))/3;
%Y Cf. A357832, A357833.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Oct 14 2022
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