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A284859
Row sums of the Sheffer triangle (exp(x), exp(3*x)-1) given in A282629.
11
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.
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. (DobiƄski 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