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
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 02 2004
EXTENSIONS
More terms from Seiichi Manyama, Jun 12 2020
STATUS
approved