A356361 a(n) = Sum_{k=0..floor(n/3)} n^k * |Stirling1(n,3*k)|.
1, 0, 0, 3, 24, 175, 1386, 12397, 125664, 1431261, 18099300, 251194911, 3788383248, 61584927495, 1072118178768, 19882255276485, 391068812992512, 8128569896422821, 177984169080865992, 4094103029211918567, 98692513234032009600, 2487731188418039207007
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Programs
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PARI
a(n) = sum(k=0, n\3, n^k*abs(stirling(n, 3*k, 1)));
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PARI
a(n) = n!*polcoef(sum(k=0, n\3, n^k*(-log(1-x+x*O(x^n)))^(3*k)/(3*k)!), n);
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PARI
Pochhammer(x, n) = prod(k=0, n-1, x+k); 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;
Formula
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)).
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.