A094420 Generalized ordered Bell numbers Bo(n,n).

1, 1, 10, 219, 8676, 544505, 49729758, 6232661239, 1026912225160, 215270320769109, 55954905981282210, 17662898483917308083, 6655958151527584785900, 2951503248457748982755953, 1521436331153097968932487206, 902143190212525713006814917615, 609729139653483641913607434550800

Offset: 0

Author

Ralf Stephan, May 02 2004

Keywords

• nonn

Comments

Main diagonal of array A094416.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A094416, A321189.

The coefficients of the Fubini polynomials are A131689.

Central column of A344499.

Programs

Magma

A094420:= func< n | (&+[Factorial(k)*n^k*StirlingSecond(n,k): k in [0..n]]) >;
[A094420(n): n in [0..25]]; // G. C. Greubel, Jan 12 2024

Maple

F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k=1..n)) end:
a := n -> subs(x = n, F(n)):
seq(a(n), n = 0..16); # Peter Luschny, May 21 2021

Mathematica

Table[Sum[k!*n^k*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jul 23 2018 *)

PARI

{a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jun 12 2020

SageMath

def aList(len):
    R. = PowerSeriesRing(QQ)
    f = lambda n: R(1/(1 + n * (1 - exp(x))))
    return [factorial(n)*f(n).list()[n] for n in (0..len-1)]
print(aList(17)) # Peter Luschny, May 21 2021

Formula

a(n) ~ sqrt(2*Pi) * n^(2*n + 5/2) / exp(n - 3/2). - Vaclav Kotesovec, Jul 23 2018

a(n) = Sum_{k=0..n} k!*n^k*Stirling2(n, k). - Seiichi Manyama, Jun 12 2020

From Peter Luschny, May 21 2021: (Start)

a(n) = F_{n}(n), the Fubini polynomial F_{n}(x) evaluated at x = n.

a(n) = n! * [x^n] (1 / (1 + n * (1 - exp(x)))). (End)

Extensions

More terms from Seiichi Manyama, Jun 12 2020