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A222079 O.g.f.: Sum_{n>=0} n^n*(n+5)^n * exp(-n*(n+5)*x) * x^n / n!.

%I A222079 #16 Feb 04 2017 21:54:50
%S A222079 1,6,62,1040,24076,703800,24786512,1020779520,48130232528,
%T A222079 2557117300640,151180506557280,9846055638729216,700523098562671360,
%U A222079 54066239308284456960,4499576117943522662400,401709919258066014720000,38299206898825369235170560,3883999501445283274005895680
%N A222079 O.g.f.: Sum_{n>=0} n^n*(n+5)^n * exp(-n*(n+5)*x) * x^n / n!.
%C A222079 From _Vaclav Kotesovec_, May 22 2014: (Start)
%C A222079 Generally, for p>=1, a(n) = 1/n!*Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+p)^n = Sum_{j=0..n} binomial(n,j) * p^(n-j) * StirlingS2(n+j,n).
%C A222079 a(n) ~ n^n * 2^(2*n+p/2) / (sqrt(2*Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+p/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599...
%C A222079 (End)
%H A222079 G. C. Greubel, <a href="/A222079/b222079.txt">Table of n, a(n) for n = 0..340</a>
%F A222079 a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+5)^k * x^k / (1 + k*(k+5)*x)^(k+1).
%F A222079 a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+5)^n.
%F A222079 a(n) ~ n^n * 2^(2*n+2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+5/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014
%e A222079 O.g.f.: A(x) = 1 + 6*x + 62*x^2 + 1040*x^3 + 24076*x^4 + 703800*x^5 +...
%e A222079 where
%e A222079 A(x) = 1 + 6*x*exp(-6*x) + 14^2*exp(-14*x)*x^2/2! + 24^3*exp(-24*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 50^5*exp(-50*x)*x^5/5! +...
%e A222079 is a power series in x with integer coefficients.
%t A222079 Flatten[{1,Table[Sum[Binomial[n,j] * 5^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)
%o A222079 (PARI) {a(n)=polcoeff(sum(m=0, n, m^m*(m+5)^m*x^m*exp(-m*(m+5)*x+x*O(x^n))/m!), n)}
%o A222079 for(n=0, 20, print1(a(n), ", "))
%o A222079 (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+5)^k*x^k/(1+k*(k+5)*x +x*O(x^n))^(k+1)), n)}
%o A222079 for(n=0, 20, print1(a(n), ", "))
%o A222079 (PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+5)^n)}
%o A222079 for(n=0, 20, print1(a(n), ", "))
%Y A222079 Cf. A217904, A217900, A222076, A222077, A222078.
%K A222079 nonn
%O A222079 0,2
%A A222079 _Paul D. Hanna_, Feb 24 2013