OFFSET
0,2
COMMENTS
FORMULA
a(n) = Sum_{m=0..n} A282629(n, m).
E.g.f.: exp(x)*exp(exp(3*x) -1).
a(n) = (1/e)*Sum_{m>=0} (1/m!)*(1+3*m)^n, n >= 0. (DobiĆski type formula from the A282629(n,m) sum formula, interchanging summations).
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - Ilya Gutkovskiy, Jun 21 2022
a(n) ~ Bell(n) * (3 + LambertW(n)/n)^n. - Vaclav Kotesovec, Jun 22 2022
a(n) ~ 3^n * n^(n + 1/3) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2022
MATHEMATICA
T[n_, m_]:= Sum[Binomial[m, k] (-1)^(k - m) (1 + 3k)^n/m!, {k, 0, m}]; Table[Sum[T[n, m], {m, 0, n}], {n, 0, 20}] (* Indranil Ghosh, Apr 10 2017 *)
Table[Sum[3^k*Binomial[n, k]*BellB[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 22 2022 *)
PROG
(PARI) T(n, m) = sum(k=0, m, binomial(m, k) * (-1)^(k - m) * (1 + 3*k)^n/m!);
a(n) = sum(m=0, n, T(n, m)); \\ Indranil Ghosh, Apr 10 2017
(Python)
from sympy import binomial, factorial
def T(n, m): return sum([binomial(m, k) * (-1)**(k - m) * (1 + 3*k)**n for k in range(m + 1)])//factorial(m)
def a(n): return sum([T(n, k) for k in range(n + 1)])
print([a(n) for n in range(20)]) # Indranil Ghosh, Apr 10 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 05 2017
STATUS
approved