A284864 Row sums of Sheffer triangle S2[3,2] given by A225466.

1, 5, 34, 287, 2839, 31898, 399079, 5480609, 81724300, 1311990425, 22521232003, 411039834356, 7938680371957, 161596770440945, 3454818798460858, 77340712165173755, 1808096791948934755, 44038966942707463946, 1115155787752634260483, 29301563957596710001709

Offset

0,2

Comments

This is a generalization of the Bell sequence A000110 because S2[3,2] is a generalization of the Stirling2 triangle A048993.

For the alternating row sums see A284865.

Links

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

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, (Dobiński 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

Cf. A000110, A225466, A284865, A284859 (case [3,1]).

Sequence in context: A111557 A211794 A289147 * A208677 A259906 A355258

Adjacent sequences: A284861 A284862 A284863 * A284865 A284866 A284867

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 10 2017

Status

approved