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%I #12 Feb 04 2017 18:14:41
%S 1,3,23,320,6397,166467,5338412,203578776,9001795829,452924585465,
%T 25555585227999,1598279794889076,109748572718377660,
%U 8209004345714098500,664396187060996529528,57853075421585981420208,5393119810256349152565573,535908449308064099732283429,56548822143306498413322880709
%N O.g.f.: Sum_{n>=0} n^n*(n+2)^n * exp(-n*(n+2)*x) * x^n / n!.
%H G. C. Greubel, <a href="/A222076/b222076.txt">Table of n, a(n) for n = 0..345</a>
%F a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+2)^k * x^k / (1 + k*(k+2)*x)^(k+1).
%F a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+2)^n.
%F a(n) ~ n^n * 2^(2*n+1/2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014
%e O.g.f.: A(x) = 1 + 3*x + 23*x^2 + 320*x^3 + 6397*x^4 + 166467*x^5 +...
%e where
%e A(x) = 1 + 3*x*exp(-3*x) + 8^2*exp(-8*x)*x^2/2! + 15^3*exp(-15*x)*x^3/3! + 24^4*exp(-24*x)*x^4/4! + 35^5*exp(-35*x)*x^5/5! +...
%e is a power series in x with integer coefficients.
%t Flatten[{1,Table[Sum[Binomial[n,j] * 2^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)
%o (PARI) {a(n)=polcoeff(sum(m=0, n, m^m*(m+2)^m*x^m*exp(-m*(m+2)*x+x*O(x^n))/m!), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+2)^k*x^k/(1+k*(k+2)*x +x*O(x^n))^(k+1)), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+2)^n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A217901, A217900, A222077, A222078, A222079.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 24 2013
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