A285064 Row sums of Sheffer triangle S2[4,1] = A285061.

1, 5, 41, 429, 5329, 75989, 1215481, 21453693, 412820385, 8579772325, 191166679497, 4538638641997, 114238219541617, 3035305413035125, 84819458105387417, 2484842038066995485, 76101249873390595905, 2430497813260105226053, 80769536433102942870377, 2787318255464814752951533

Offset

0,2

Comments

See A285061 for details. These are generalized Bell numbers (A000110) because A285061 is a generalized Stirling2 triangle.

For the alternating row sums of A285061 see A285065.

Links

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

Formula

a(n) = Sum_{m=0..n} A285061(n, m), n >= 0.

E.g.f.: exp(x)*exp(exp(4*x) - 1).

a(n) = (1/e)*Sum_{m>=0} (1/m!)*(1+4*m)^n, n >= 0. (Dobiński type formula from the A285061(n,m) sum formula, after interchange of summations).

a(n) = Sum_{k=0..n} binomial(n, k)*A000110(k)*4^k, n >= 0. From the Vaclav Kotesovec program. This follows from the S2[4,1] formula in terms of Stirling2. - Wolfdieter Lang, Apr 24 2017

a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k). - Ilya Gutkovskiy, Jun 21 2022

a(n) ~ Bell(n) * (4 + LambertW(n)/n)^n. - Vaclav Kotesovec, Jun 22 2022

a(n) ~ 4^n * n^(n + 1/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/4)). - Vaclav Kotesovec, Jun 27 2022

Mathematica

Table[Sum[Binomial[n, k]*BellB[k]*4^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2017 *)

Prog

(Python)
from sympy import binomial, bell
def a(n): return sum([binomial(n, k)*bell(k)*4**k for k in range(n + 1)]) # Indranil Ghosh, Apr 19 2017

Crossrefs

Cf. A000110, A285061, A285065.

Cf. A126390, A284859.

Sequence in context: A177506 A064087 A329123 * A232685 A081215 A218219

Adjacent sequences: A285061 A285062 A285063 * A285065 A285066 A285067

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 13 2017

Status

approved