A094420 - OEIS

%I #31 Jan 12 2024 01:15:58

%S 1,1,10,219,8676,544505,49729758,6232661239,1026912225160,

%T 215270320769109,55954905981282210,17662898483917308083,

%U 6655958151527584785900,2951503248457748982755953,1521436331153097968932487206,902143190212525713006814917615,609729139653483641913607434550800

%N Generalized ordered Bell numbers Bo(n,n).

%C Main diagonal of array A094416.

%H Seiichi Manyama, <a href="/A094420/b094420.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ sqrt(2*Pi) * n^(2*n + 5/2) / exp(n - 3/2). - _Vaclav Kotesovec_, Jul 23 2018

%F a(n) = Sum_{k=0..n} k!*n^k*Stirling2(n, k). - _Seiichi Manyama_, Jun 12 2020

%F From _Peter Luschny_, May 21 2021: (Start)

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

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

%p F := proc(n) option remember; if n = 0 then return 1 fi;

%p expand(add(binomial(n, k)*F(n-k)*x, k=1..n)) end:

%p a := n -> subs(x = n, F(n)):

%p seq(a(n), n = 0..16); # _Peter Luschny_, May 21 2021

%t Table[Sum[k!*n^k*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 23 2018 *)

%o (PARI) {a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 2))} \\ _Seiichi Manyama_, Jun 12 2020

%o (SageMath)

%o def aList(len):

%o     R.<x> = PowerSeriesRing(QQ)

%o     f = lambda n: R(1/(1 + n * (1 - exp(x))))

%o     return [factorial(n)*f(n).list()[n] for n in (0..len-1)]

%o print(aList(17)) # _Peter Luschny_, May 21 2021

%o (Magma)

%o A094420:= func< n | (&+[Factorial(k)*n^k*StirlingSecond(n,k): k in [0..n]]) >;

%o [A094420(n): n in [0..25]]; // _G. C. Greubel_, Jan 12 2024

%Y Cf. A094416, A321189.

%Y The coefficients of the Fubini polynomials are A131689.

%Y Central column of A344499.

%K nonn

%O 0,3

%A _Ralf Stephan_, May 02 2004

%E More terms from _Seiichi Manyama_, Jun 12 2020