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

0,4

Links

Table of n, a(n) for n=0..21.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

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.

Prog

(PARI) a(n) = sum(k=0, n\3, n^k*abs(stirling(n, 3*k, 1)));
(PARI) a(n) = n!*polcoef(sum(k=0, n\3, n^k*(-log(1-x+x*O(x^n)))^(3*k)/(3*k)!), n);
(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;

Crossrefs

Cf. A000407, A357683.

Cf. A356362, A356363.

Cf. A079978, A357831.

Sequence in context: A120741 A361553 A356362 * A292293 A073985 A197209

Adjacent sequences: A356358 A356359 A356360 * A356362 A356363 A356364

Keyword

nonn

Author

Seiichi Manyama, Oct 16 2022

Status

approved