OFFSET
0,2
COMMENTS
FORMULA
a(n) = Sum_{k=0..n} A225466(n, k), n >= 0.
E.g.f.: exp(2*x)*exp((exp(3*x)-1)) (Sheffer property).
a(n) = (1/e)*Sum_{m>=0} (1/m!)*(2 + 3*m)^n, n >= 0, (DobiĆski type formula).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - Ilya Gutkovskiy, Jun 21 2022
a(n) ~ 3^n * n^(n + 2/3) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 2/3)). - Vaclav Kotesovec, Jun 27 2022
MATHEMATICA
T[n_, k_]:=Sum[Binomial[k, j](-1)^(j - k) (2 + 3j)^n/k!, {j, 0, k}]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 20}] (* Indranil Ghosh, Apr 10 2017 *)
PROG
(PARI) T(n, k) = sum(j=0, k, binomial(k, j)*(-1)^(j - k)*(2 + 3*j)^n/k!);
a(n) = sum(k=0, n, T(n, k)); \\ Indranil Ghosh, Apr 10 2017
(Python)
from sympy import binomial, factorial
def T(n, k): return sum([binomial(k, j)*(-1)**(j - k)*(2 + 3*j)**n/factorial(k) for j in range(k + 1)])
def a(n): return sum([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 10 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 10 2017
STATUS
approved