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

Offset

0,2

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.

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

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. *)

Prog

(Magma) m:=20; R<x>:=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
(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
(PARI) my(N=25, x='x+O('x^N)); Vec(serlaplace(1/(5 - 4*exp(x)))) \\ Joerg Arndt, Jan 15 2024

Crossrefs

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

Cf. A346983, A354242, A365567.

Sequence in context: A291313 A002690 A370927 * A349504 A354264 A138435

Adjacent sequences: A094414 A094415 A094416 * A094418 A094419 A094420

Keyword

nonn

Author

Ralf Stephan, May 02 2004

Status

approved