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

0,2

Comments

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

Links

Table of n, a(n) for n=0..19.

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

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

Cf. A000110, A126390, A282629, A284864.

Sequence in context: A327074 A337167 A140094 * A144647 A215505 A182304

Adjacent sequences: A284856 A284857 A284858 * A284860 A284861 A284862

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 05 2017

Status

approved