A094417 Generalized ordered Bell numbers Bo(4,n).

1, 4, 36, 484, 8676, 194404, 5227236, 163978084, 5878837476, 237109864804, 10625889182436, 523809809059684, 28168941794178276, 1641079211868751204, 102961115527874385636, 6921180217049667005284, 496267460209336700111076, 37807710659221213027893604

Offset: 0

Author

Ralf Stephan, May 02 2004

Keywords

• nonn

Comments

Fourth row of array A094416, which has more information.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Crossrefs

Cf. A000670, A004123, A032033, A050353, A094418, A094419, A131689.

Cf. A346983, A354242, A365567.

Programs

Magma

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(5 - 4*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Bruno Berselli, Mar 17 2014

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, 4* add(binomial(n, k) *a(k), k=0..n-1))
    end:
seq(a(n), n=0..20);

Mathematica

max = 16; f[x_] := 1/(5-4*E^x); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 14 2011, after g.f. *)

PARI

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(5 - 4*exp(x)))) \\ Joerg Arndt, Jan 15 2024

SageMath

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094417(k): return A094416(4,k)
[A094417(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

Formula

E.g.f.: 1/(5 - 4*exp(x)).

a(n) = 4 * A050353(n) for n>0.

a(n) = Sum_{k=0..n} A131689(n,k) * 4^k. - Philippe Deléham, Nov 03 2008

E.g.f.: A(x) with A_n = 4 * Sum_{k=0..n-1} C(n,k) * A_k; A_0 = 1. - Vladimir Kruchinin, Jan 27 2011

G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 10*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

a(n) = log(5/4)*int {x = 0..inf} (floor(x))^n * (5/4)^(-x) dx. - Peter Bala, Feb 14 2015

a(0) = 1; a(n) = 4 * a(n-1) - 5 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

From Seiichi Manyama, Jun 01 2025: (Start)

a(n) = (-1)^(n+1)/5 * Li_{-n}(5/4), where Li_{n}(x) is the polylogarithm function.

a(n) = (1/5) * Sum_{k>=0} k^n * (4/5)^k.

a(n) = (4/5) * Sum_{k=0..n} 5^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)