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%I #13 Feb 04 2017 19:12:46
%S 1,4,34,504,10572,285408,9419440,367571200,16562241744,846509123520,
%T 48401180913824,3061687935718272,212316590908782336,
%U 16018267935253721088,1306322033185206970368,114519518777575592865792,10740222055670467832259840,1073051903942317493993088000
%N O.g.f.: Sum_{n>=0} n^n*(n+3)^n * exp(-n*(n+3)*x) * x^n / n!.
%H G. C. Greubel, <a href="/A222077/b222077.txt">Table of n, a(n) for n = 0..340</a>
%F a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+3)^k * x^k / (1 + k*(k+3)*x)^(k+1).
%F a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+3)^n.
%F a(n) ~ n^n * 2^(2*n+1) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+3/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014
%e O.g.f.: A(x) = 1 + 4*x + 34*x^2 + 504*x^3 + 10572*x^4 + 285408*x^5 +...
%e where
%e A(x) = 1 + 4*x*exp(-4*x) + 10^2*exp(-10*x)*x^2/2! + 18^3*exp(-18*x)*x^3/3! + 28^4*exp(-28*x)*x^4/4! + 40^5*exp(-40*x)*x^5/5! +...
%e is a power series in x with integer coefficients.
%t Flatten[{1,Table[Sum[Binomial[n,j] * 3^(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+3)^m*x^m*exp(-m*(m+3)*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+3)^k*x^k/(1+k*(k+3)*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+3)^n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A217902, A217900, A222076, A222078, A222079.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 24 2013
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