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a(n) = Sum_{k=0..floor((n-2)/3)} Stirling1(n,3*k+2).
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%I #12 Oct 15 2022 08:08:33

%S 0,0,1,-3,11,-49,259,-1589,11109,-87171,758121,-7229859,74905467,

%T -836159961,9980000667,-126422745813,1686902233653,-23512989735963,

%U 338917341235473,-4982536435536387,73087736506615467,-1025163078325255233,12286912220375608179

%N a(n) = Sum_{k=0..floor((n-2)/3)} Stirling1(n,3*k+2).

%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*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(log(1+x)).

%F a(n) = (-1)^n * ( (-1)_n + w * (-w)_n + w^2 * (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.

%o (PARI) a(n) = sum(k=0, (n-2)\3, stirling(n, 3*k+2, 1));

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k+2)/(3*k+2)!))))

%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*Pochhammer(-w, n)+w^2*Pochhammer(-w^2, n))/3;

%Y Cf. A357834, A357835.

%K sign

%O 0,4

%A _Seiichi Manyama_, Oct 14 2022