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A222078 O.g.f.: Sum_{n>=0} n^n*(n+4)^n * exp(-n*(n+4)*x) * x^n / n!.

%I A222078 #14 Feb 04 2017 21:53:09
%S A222078 1,5,47,742,16357,459369,15651260,626935936,28872594389,1503262704775,
%T A222078 87328047029511,5600639046765690,393092088068927860,
%U A222078 29974039720132943036,2467669218502361588472,218168186315818183909344,20617367868151866462395205,2074120178028300166990286691
%N A222078 O.g.f.: Sum_{n>=0} n^n*(n+4)^n * exp(-n*(n+4)*x) * x^n / n!.
%H A222078 G. C. Greubel, <a href="/A222078/b222078.txt">Table of n, a(n) for n = 0..340</a>
%F A222078 a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+4)^k * x^k / (1 + k*(k+4)*x)^(k+1).
%F A222078 a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+4)^n.
%F A222078 a(n) ~ n^n * 2^(2*n+3/2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014
%e A222078 O.g.f.: A(x) = 1 + 5*x + 47*x^2 + 742*x^3 + 16357*x^4 + 459369*x^5 +...
%e A222078 where
%e A222078 A(x) = 1 + 5*x*exp(-5*x) + 12^2*exp(-12*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 32^4*exp(-32*x)*x^4/4! + 45^5*exp(-45*x)*x^5/5! +...
%e A222078 is a power series in x with integer coefficients.
%t A222078 Flatten[{1,Table[Sum[Binomial[n,j] * 4^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)
%o A222078 (PARI) {a(n)=polcoeff(sum(m=0, n, m^m*(m+4)^m*x^m*exp(-m*(m+4)*x+x*O(x^n))/m!), n)}
%o A222078 for(n=0, 20, print1(a(n), ", "))
%o A222078 (PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+4)^k*x^k/(1+k*(k+4)*x +x*O(x^n))^(k+1)), n)}
%o A222078 for(n=0, 20, print1(a(n), ", "))
%o A222078 (PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+4)^n)}
%o A222078 for(n=0, 20, print1(a(n), ", "))
%Y A222078 Cf. A217903, A217900, A222076, A222077, A222079.
%K A222078 nonn
%O A222078 0,2
%A A222078 _Paul D. Hanna_, Feb 24 2013