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A356361 a(n) = Sum_{k=0..floor(n/3)} n^k * |Stirling1(n,3*k)|.

%I A356361 #16 Feb 16 2025 08:34:03
%S A356361 1,0,0,3,24,175,1386,12397,125664,1431261,18099300,251194911,
%T A356361 3788383248,61584927495,1072118178768,19882255276485,391068812992512,
%U A356361 8128569896422821,177984169080865992,4094103029211918567,98692513234032009600,2487731188418039207007
%N A356361 a(n) = Sum_{k=0..floor(n/3)} n^k * |Stirling1(n,3*k)|.
%H A356361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A356361 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! + ... . a(n) = n! * [x^n] F(-n^(1/3) * log(1-x)).
%F A356361 a(n) = ( (n^(1/3))_n + (n^(1/3)*w)_n + (n^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.
%o A356361 (PARI) a(n) = sum(k=0, n\3, n^k*abs(stirling(n, 3*k, 1)));
%o A356361 (PARI) a(n) = n!*polcoef(sum(k=0, n\3, n^k*(-log(1-x+x*O(x^n)))^(3*k)/(3*k)!), n);
%o A356361 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A356361 a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); round(Pochhammer(v, n)+Pochhammer(v*w, n)+Pochhammer(v*w^2, n))/3;
%Y A356361 Cf. A000407, A357683.
%Y A356361 Cf. A356362, A356363.
%Y A356361 Cf. A079978, A357831.
%K A356361 nonn
%O A356361 0,4
%A A356361 _Seiichi Manyama_, Oct 16 2022