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A218670 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^n * x^n/n! * exp(-n*x*(1+n*x)).

%I A218670 #11 May 10 2014 03:03:52
%S A218670 1,1,2,7,26,116,556,2927,16388,97666,612136,4023878,27579410,
%T A218670 196537134,1451102836,11074811191,87160086800,706055915318,
%U A218670 5876662642720,50182337830986,439036984440316,3930618736372336,35970734643745496,336153100655220126,3205000520319374116
%N A218670 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^n * x^n/n! * exp(-n*x*(1+n*x)).
%C A218670 Compare the o.g.f. to the curious identity:
%C A218670 1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).
%H A218670 Vaclav Kotesovec, <a href="/A218670/b218670.txt">Table of n, a(n) for n = 0..350</a>
%e A218670 O.g.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 116*x^5 + 556*x^6 + 2927*x^7 +...
%e A218670 where
%e A218670 A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^2*(1+2*x)^2*x^2/2!*exp(-2*x*(1+2*x)) + 3^3*(1+3*x)^3*x^3/3!*exp(-3*x*(1+3*x)) + 4^4*(1+4*x)^4*x^4/4!*exp(-4*x*(1+4*x)) + 5^5*(1+5*x)^5*x^5/5!*exp(-5*x*(1+5*x)) +...
%e A218670 simplifies to a power series in x with integer coefficients.
%o A218670 (PARI) {a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^k*x^k/k!*exp(-k*x*(1+k*x)+x*O(x^n)));polcoeff(A,n)}
%o A218670 for(n=0,30,print1(a(n),", "))
%Y A218670 Cf. A218677, A218678, A218679, A218667, A218668, A218669, A134055, A218671, A217900.
%K A218670 nonn
%O A218670 0,3
%A A218670 _Paul D. Hanna_, Nov 04 2012