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O.g.f.: Sum_{n>=0} x^n * (n + x)^n / (1 + n*x)^n.
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%I #7 Oct 30 2014 17:17:59

%S 1,1,4,15,73,425,2908,22855,202849,2005929,21864076,260374631,

%T 3363097609,46823803585,699001981588,11137295369775,188636060894593,

%U 3384325253935025,64113731067110644,1278893092183672159,26792685755013073801,588160948075800731961,13500922657476722741164

%N O.g.f.: Sum_{n>=0} x^n * (n + x)^n / (1 + n*x)^n.

%C Compare to the identity:

%C Sum_{n>=0} n^n * x^n / (1 + n*x)^n = 1/2 + Sum_{n>=0} (n+1)!/2 * x^n.

%H Vaclav Kotesovec, <a href="/A230741/b230741.txt">Table of n, a(n) for n = 0..400</a>

%e G.f.: A(x) = 1 + x + 4*x^2 + 15*x^3 + 73*x^4 + 425*x^5 + 2908*x^6 +...

%e where

%e A(x) = 1 + x*(1+x)/(1+x) + x^2*(2+x)^2/(1+2*x)^2 + x^3*(3+x)^3/(1+3*x)^3 + x^4*(4+x)^4/(1+4*x)^4 + x^5*(5+x)^5/(1+5*x)^5 +...

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

%o for(n=0, 30, print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 28 2013