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

0,3

Comments

Main diagonal of array A094416.

Links

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

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)

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 *)

Prog

(PARI) {a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jun 12 2020
(SageMath)
def aList(len):
    R.<x> = 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
(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

Crossrefs

Cf. A094416, A321189.

The coefficients of the Fubini polynomials are A131689.

Central column of A344499.

Sequence in context: A007699 A024291 A024292 * A053131 A334908 A166181

Adjacent sequences: A094417 A094418 A094419 * A094421 A094422 A094423

Keyword

nonn

Author

Ralf Stephan, May 02 2004

Extensions

More terms from Seiichi Manyama, Jun 12 2020

Status

approved