A284859 Row sums of the Sheffer triangle (exp(x), exp(3*x)-1) given in A282629.

1, 4, 25, 199, 1876, 20257, 245017, 3266914, 47450923, 743935375, 12497579698, 223619318215, 4240423494685, 84855613320004, 1785410320771933, 39373503608087299, 907548770965519660, 21810536356271794549, 545305573054110017125, 14155835044848094831018

Offset: 0

Author

Wolfdieter Lang, Apr 05 2017

Keywords

• nonn
• easy

Comments

See A282629 for details. These are generalized Bell numbers (A000110) because A282629 is a generalized Stirling2 triangle.

Crossrefs

Cf. A000110, A126390, A282629, A284864.

Programs

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 *)

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

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. (Dobiński 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