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%I #14 Oct 16 2022 07:59:27
%S 1,0,0,3,-24,175,-1314,10339,-84448,696429,-5444700,32897601,53444304,
%T -8071238721,235927045536,-5630771421765,126525509087232,
%U -2799633511755963,62154971516786616,-1396560425289392007,31880150704745078400,-740188445913015688953
%N a(n) = Sum_{k=0..floor(n/3)} n^k * Stirling1(n,3*k).
%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) + 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 a(n) = (-1)^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 (PARI) a(n) = sum(k=0, n\3, n^k*stirling(n, 3*k, 1));
%o (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 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-v, n)+Pochhammer(-v*w, n)+Pochhammer(-v*w^2, n))/3;
%Y Cf. A356361, A356363.
%Y Cf. A357834.
%K sign
%O 0,4
%A _Seiichi Manyama_, Oct 16 2022
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