%I #13 Oct 15 2022 08:08:37
%S 0,1,-1,2,-5,14,-35,-14,1701,-26418,351351,-4622982,62705643,
%T -890078826,13297263525,-209438953542,3477446002485,-60803484275898,
%U 1117975706702127,-21580455768575886,436591651807054107,-9241512424454751714,204338436416329792941
%N a(n) = Sum_{k=0..floor((n-1)/3)} Stirling1(n,3*k+1).
%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) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(log(1+x)).
%F a(n) = (-1)^n * ( (-1)_n + w^2 * (-w)_n + w * (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.
%o (PARI) a(n) = sum(k=0, (n-1)\3, stirling(n, 3*k+1, 1));
%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k+1)/(3*k+1)!))))
%o (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+w^2*Pochhammer(-w, n)+w*Pochhammer(-w^2, n))/3;
%Y Cf. A357834, A357836.
%K sign
%O 0,4
%A _Seiichi Manyama_, Oct 14 2022
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