A356363 a(n) = Sum_{k=0..floor(n/3)} n^k * Stirling2(n,3*k).

1, 0, 0, 3, 24, 125, 576, 3136, 24752, 242280, 2421000, 23568743, 230156136, 2370756505, 26664718080, 326641069815, 4243004068192, 57065900282730, 787656999701016, 11193821784313606, 165023822310642520, 2535785869709189307, 40583218821499596176

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Eric Weisstein's MathWorld, Bell Polynomial.

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) * (exp(x)-1)).

a(n) = ( Bell_n(n^(1/3)) + Bell_n(n^(1/3)*w) + Bell_n(n^(1/3)*w^2) )/3, where Bell_n(x) is n-th Bell polynomial.

Prog

(PARI) a(n) = sum(k=0, n\3, n^k*stirling(n, 3*k, 2));
(PARI) a(n) = n!*polcoef(sum(k=0, n\3, n^k*(exp(x+x*O(x^n))-1)^(3*k)/(3*k)!), n);
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, v)+Bell_poly(n, v*w)+Bell_poly(n, v*w^2))/3;

Crossrefs

Cf. A356361, A356362.

Cf. A357782.

Sequence in context: A319097 A326789 A305543 * A183900 A001089 A359884

Adjacent sequences: A356360 A356361 A356362 * A356364 A356365 A356366

Keyword

nonn

Author

Seiichi Manyama, Oct 16 2022

Status

approved