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A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).

%I A121886 #21 Sep 07 2018 21:24:36
%S A121886 1,1,5,40,444,6324,110023,2261576,53632424,1441341350,43290170494,
%T A121886 1437020742408,52243864528990,2064488610832106,88106523694973953,
%U A121886 4038627301344466648,197888243609535940091,10321811633042512528240
%N A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).
%C A121886 Number of square matrices with nonnegative integer entries and without zero rows such that sum of all entries is equal to n. - _Vladeta Jovovic_, Mar 04 2008
%F A121886 G.f.: Sum_{n>=0} ( 1/(1-x)^n - 1 )^n.
%F A121886 G.f.: Sum_{n>=0} (1-x)^n / (1 + (1-x)^n)^(n+1). - _Paul D. Hanna_, Sep 07 2018
%F A121886 a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.38377369607518184186200387319561108... . - _Vaclav Kotesovec_, May 07 2014
%e A121886 G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 444*x^4 + 6324*x^5 +...
%e A121886 where
%e A121886 A(x) = 1 + (1/(1-x) - 1) + (1/(1-x)^2 - 1)^2 + (1/(1-x)^3 - 1)^3 + ...
%e A121886 Also,
%e A121886 A(x) = 1/2 + (1-x)/(1 + (1-x))^2 + (1-x)^2/(1 + (1-x)^2)^3 +  + (1-x)^3/(1 + (1-x)^3)^4 + (1-x)^4/(1 + (1-x)^4)^5 + ...
%t A121886 Flatten[{1,Table[1/n!* Sum[Abs[StirlingS1[n,k]]*Sum[m^k*m!*StirlingS2[k, m], {m, 1, k}],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 07 2014 *)
%o A121886 (PARI) {a(n)=polcoeff(sum(m=0,n,(1/(1-x+x*O(x^n))^m-1)^m),n)}
%Y A121886 Cf. A104209, A220352.
%K A121886 easy,nonn
%O A121886 0,3
%A A121886 _Vladeta Jovovic_, Aug 31 2006
%E A121886 More terms from _Max Alekseyev_, Feb 01 2007