A094418 Generalized ordered Bell numbers Bo(5,n).

1, 5, 55, 905, 19855, 544505, 17919055, 687978905, 30187495855, 1490155456505, 81732269223055, 4931150091426905, 324557348772511855, 23141780973332248505, 1776997406800302687055, 146197529083891406394905

Offset

0,2

Comments

Fifth 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/(6 - 5*exp(x)).

a(n) = Sum_{k=0..n} A131689(n,k)*5^k. - Philippe Deléham, Nov 03 2008

a(n) ~ n! / (6*(log(6/5))^(n+1)). - Vaclav Kotesovec, Mar 14 2014

a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020

a(0) = 1; a(n) = 5*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

Mathematica

t = 30; Range[0, t]! CoefficientList[Series[1/(6 - 5 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

Prog

(Magma)
A094416:= func< n, k | (&+[Factorial(j)*n^j*StirlingSecond(k, j): j in [0..k]]) >;
A094418:= func< k | A094416(5, k) >;
[A094418(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024
(SageMath)
def A094416(n, k): return sum(factorial(j)*n^j*stirling_number2(k, j) for j in range(k+1)) # array
def A094418(k): return A094416(5, k)
[A094418(n) for n in range(31)] # G. C. Greubel, Jan 12 2024
(PARI) my(N=25, x='x+O('x^N)); Vec(serlaplace(1/(6 - 5*exp(x)))) \\ Joerg Arndt, Jan 15 2024

Crossrefs

Cf. A094416, A094417, A094419, A094422, A131689.

Cf. A346984, A365568, A365569, A365570.

Sequence in context: A294051 A145662 A362653 * A365588 A367164 A008543

Adjacent sequences: A094415 A094416 A094417 * A094419 A094420 A094421

Keyword

nonn

Author

Ralf Stephan, May 02 2004

Status

approved