OFFSET
0,4
LINKS
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 16 2022
STATUS
approved