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%I #23 Nov 29 2022 12:27:00
%S 1,1,5,44,548,8808,173352,4036288,108507968,3307368320,112703108480,
%T 4245680193024,175200825481728,7859411394860032,380810598813553664,
%U 19819617775693512704,1102737068471914938368,65316500202537025634304,4103422475123595857854464
%N Expansion of Sum_{n>=0} n^n*x^n/(1 - n*x)^n.
%C Compare g.f. to the identity (cf. A001710):
%C Sum_{n>=0} n^n*x^n/(1 + n*x)^n = 1 + (1/2)*Sum_{n>=1} (n+1)!*x^n.
%H Seiichi Manyama, <a href="/A195242/b195242.txt">Table of n, a(n) for n = 0..371</a>
%F a(n) = Sum_{k=0..n} C(n-1,k)*(k+1)^n.
%F a(n) = (n+1)!/2 + 2*Sum_{k=0..[n/2]} C(n-1,n-2*k)*(n-2*k+1)^n for n>0 with a(0)=1.
%F a(n) ~ n^n * r^(n+3/2) / (exp(n) * (1-r)^n), where r = 1/(1+LambertW(exp(-1))) = 0.78218829428019990122... . - _Vaclav Kotesovec_, May 14 2014
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(-k,-n)*k^n. Cf. A053506. - _Peter Luschny_, Apr 11 2016
%e G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 548*x^4 + 8808*x^5 + 173352*x^6 +...
%e where
%e A(x) = 1 + x/(1-x) + 2^2*x^2/(1-2*x)^2 + 3^3*x^3/(1-3*x)^3 + 4^4*x^4/(1-4*x)^4 +...
%t a[n_] := Sum[Binomial[n - 1, k] (k + 1)^n, {k, 0, n}];
%t Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jun 26 2019 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,m^m*x^m/(1-m*x+x*O(x^n))^m),n)}
%o (PARI) {a(n)=sum(k=0,n,binomial(n-1,k)*(k+1)^n)}
%o (PARI) {a(n)=(n+1)!/2 + 2*sum(k=0,n\2,binomial(n-1,n-2*k)*(n-2*k+1)^n)}
%Y Cf. A001710, A053506, A242449.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 13 2011
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