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a(n) = exp(-1) * (n+1)! * Sum_{i>j>=0} j^n/i!.
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%I #28 Jun 20 2023 10:02:37

%S 1,1,5,42,544,10020,246840,7790160,305234496,14493790080,818125056000,

%T 54040182134400,4122847701941760,359308852504842240,

%U 35434445974210483200,3922092526740740352000,483768942063738966835200,66076388642496229027430400,9938208084601523840431718400

%N a(n) = exp(-1) * (n+1)! * Sum_{i>j>=0} j^n/i!.

%H Robert Israel, <a href="/A102244/b102244.txt">Table of n, a(n) for n = 0..273</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F a(n) = Sum(k=0..n, (n+1)!/(k+1)*stirling2(n,k)). - _Vladimir Reshetnikov_, Oct 20 2015

%F a(n) = (n+1)!*Integrate(x=0..1, B_n(x)), where B_n(x) is n-th Bell polynomial. - _Vladimir Reshetnikov_, Oct 21 2015

%p seq(add((n+1)!/(k+1)*Stirling2(n,k),k=0..n), n=0..20); # _Robert Israel_, Oct 21 2015

%p # Using the triangle algorithm described in A363732:

%p b := proc(n, m, x) option remember; if n = 0 then (-1)^m/(m + 1)! else

%p -(n + 1) * ((m + 1)*b(n - 1, m + 1, x) - (m + 1 - x)*b(n - 1, m, x)) fi end:

%p a := n -> local k; add(coeff(b(n, 0, x), x, k), k = 0..n):

%p seq(a(n), n = 0..18); # _Peter Luschny_, Jun 20 2023

%t Table[Sum[(n + 1)!/(k + 1) StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 20 2015 *)

%o (PARI) a(n)=round((n+1)!*exp(-1)*sum(i1=0,200, sum(i2=i1+1,200,i1^n*1./i2!)))

%Y Cf. A000110, A363732.

%K nonn

%O 0,3

%A _Benoit Cloitre_, Feb 18 2005

%E Typo in name corrected by _Vladimir Reshetnikov_, Oct 20 2015