%I #20 Oct 16 2022 16:35:11

%S 0,1,1,2,8,44,290,2194,18690,177072,1848048,21079332,260998584,

%T 3487438476,50030096844,767092681992,12520306878720,216760973139072,

%U 3967857438205320,76575231882844056,1553981718941428824,33082675130470434336,737250032464248840192

%N a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * |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(-2^(1/3) * log(1-x))/(2^(1/3)).

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

%t a[n_] := With[{v = 2^(1/3), w = (-1 + Sqrt[3]*I)/2}, Round[(Pochhammer[v, n] + w^2*Pochhammer[v*w, n] + w*Pochhammer[v*w^2, n])/(3*v)]];

%t Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Oct 16 2022, after 3rd PARI code *)

%o (PARI) a(n) = sum(k=0, (n-1)\3, 2^k*abs(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, 2^k*(-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(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Pochhammer(v, n)+w^2*Pochhammer(v*w, n)+w*Pochhammer(v*w^2, n))/(3*v));

%Y Cf. A357831, A357833.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 14 2022