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A357832 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * |Stirling1(n,3*k+1)|.

%I A357832 #21 Feb 16 2025 08:34:04
%S A357832 0,1,1,2,8,44,290,2194,18690,177072,1848048,21079332,260998584,
%T A357832 3487438476,50030096844,767092681992,12520306878720,216760973139072,
%U A357832 3967857438205320,76575231882844056,1553981718941428824,33082675130470434336,737250032464248840192
%N A357832 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * |Stirling1(n,3*k+1)|.
%H A357832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357832 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 A357832 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 A357832 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 A357832 Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Oct 16 2022, after 3rd PARI code *)
%o A357832 (PARI) a(n) = sum(k=0, (n-1)\3, 2^k*abs(stirling(n, 3*k+1, 1)));
%o A357832 (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 A357832 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A357832 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 A357832 Cf. A357831, A357833.
%K A357832 nonn
%O A357832 0,4
%A A357832 _Seiichi Manyama_, Oct 14 2022