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

%I A094420 #31 Jan 12 2024 01:15:58
%S A094420 1,1,10,219,8676,544505,49729758,6232661239,1026912225160,
%T A094420 215270320769109,55954905981282210,17662898483917308083,
%U A094420 6655958151527584785900,2951503248457748982755953,1521436331153097968932487206,902143190212525713006814917615,609729139653483641913607434550800
%N A094420 Generalized ordered Bell numbers Bo(n,n).
%C A094420 Main diagonal of array A094416.
%H A094420 Seiichi Manyama, <a href="/A094420/b094420.txt">Table of n, a(n) for n = 0..200</a>
%F A094420 a(n) ~ sqrt(2*Pi) * n^(2*n + 5/2) / exp(n - 3/2). - _Vaclav Kotesovec_, Jul 23 2018
%F A094420 a(n) = Sum_{k=0..n} k!*n^k*Stirling2(n, k). - _Seiichi Manyama_, Jun 12 2020
%F A094420 From _Peter Luschny_, May 21 2021: (Start)
%F A094420 a(n) = F_{n}(n), the Fubini polynomial F_{n}(x) evaluated at x = n.
%F A094420 a(n) = n! * [x^n] (1 / (1 + n * (1 - exp(x)))).  (End)
%p A094420 F := proc(n) option remember; if n = 0 then return 1 fi;
%p A094420 expand(add(binomial(n, k)*F(n-k)*x, k=1..n)) end:
%p A094420 a := n -> subs(x = n, F(n)):
%p A094420 seq(a(n), n = 0..16); # _Peter Luschny_, May 21 2021
%t A094420 Table[Sum[k!*n^k*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 23 2018 *)
%o A094420 (PARI) {a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 2))} \\ _Seiichi Manyama_, Jun 12 2020
%o A094420 (SageMath)
%o A094420 def aList(len):
%o A094420     R.<x> = PowerSeriesRing(QQ)
%o A094420     f = lambda n: R(1/(1 + n * (1 - exp(x))))
%o A094420     return [factorial(n)*f(n).list()[n] for n in (0..len-1)]
%o A094420 print(aList(17)) # _Peter Luschny_, May 21 2021
%o A094420 (Magma)
%o A094420 A094420:= func< n | (&+[Factorial(k)*n^k*StirlingSecond(n,k): k in [0..n]]) >;
%o A094420 [A094420(n): n in [0..25]]; // _G. C. Greubel_, Jan 12 2024
%Y A094420 Cf. A094416, A321189.
%Y A094420 The coefficients of the Fubini polynomials are A131689.
%Y A094420 Central column of A344499.
%K A094420 nonn
%O A094420 0,3
%A A094420 _Ralf Stephan_, May 02 2004
%E A094420 More terms from _Seiichi Manyama_, Jun 12 2020