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

1, 1, 5, 40, 444, 6324, 110023, 2261576, 53632424, 1441341350, 43290170494, 1437020742408, 52243864528990, 2064488610832106, 88106523694973953, 4038627301344466648, 197888243609535940091, 10321811633042512528240

Offset: 0

Author

Vladeta Jovovic, Aug 31 2006

Keywords

• easy
• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 444*x^4 + 6324*x^5 +...
where
A(x) = 1 + (1/(1-x) - 1) + (1/(1-x)^2 - 1)^2 + (1/(1-x)^3 - 1)^3 + ...
Also,
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 + ...
		

Crossrefs

Cf. A104209, A220352.

Programs

Mathematica

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 *)

PARI

{a(n)=polcoeff(sum(m=0,n,(1/(1-x+x*O(x^n))^m-1)^m),n)}

Formula

G.f.: Sum_{n>=0} ( 1/(1-x)^n - 1 )^n.

G.f.: Sum_{n>=0} (1-x)^n / (1 + (1-x)^n)^(n+1). - Paul D. Hanna, Sep 07 2018

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

Extensions

More terms from Max Alekseyev, Feb 01 2007