OFFSET
0,2
FORMULA
a(0) = 1; a(n) = 3*n*a(n-1) + Sum_{k=1..n} (-1)^(k-1) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..n} 3^(n-k) * (n-k+1)^k / k!.
MATHEMATICA
a[n_] := n! Sum[3^(n - k) (n - k + 1)^k / k!, {k, 0, n}]; Table[a[n], {n, 0, 17}] (* or *) a[0] = 1; a[n_] := 3n a[n - 1] + Sum[(-1)^(k - 1) Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}] (* James C. McMahon, Dec 18 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k+1)^k/k!);
(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(exp(-x) - 3*x))) \\ Michel Marcus, Dec 18 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Dec 18 2023
STATUS
approved