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A162697 E.g.f. satisfies: A(x) = 1+x + x^2*exp(x*A(x)). 1

%I

%S 1,1,2,6,36,260,2190,23562,294056,4145976,66518010,1187366510,

%T 23307288972,500683995396,11669239646246,293211947901810,

%U 7905976017270480,227653751742812912,6972326784534024306

%N E.g.f. satisfies: A(x) = 1+x + x^2*exp(x*A(x)).

%F a(n) = n!*Sum_{k=0..n} Sum_{j=0..k} C(j+1,n-k)/(j+1) * C(n-k,k-j)*(k-j)^j/j!.

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

%F a(n,m) = n!*Sum_{k=0..n} Sum_{j=0..k} m*C(j+m,n-k)/(j+m) * C(n-k,k-j)*(k-j)^j/j!.

%F ...

%F exp(x*A(x)) = G(x) = exp(x+x^2+x^3*G(x)) is the e.g.f. of A162161:

%F a(n) = n*(n-1)*A162161(n-2) for n>=2.

%F E.g.f.: 1 + x - LambertW(-exp(x*(1+x))*x^3)/x. - _Vaclav Kotesovec_, Feb 26 2014

%F a(n) ~ sqrt(2*r^2+r+3) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.542223654754281322169639... is the root of the equation exp(r^2+r+1)*r^3 = 1. - _Vaclav Kotesovec_, Feb 26 2014

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! +...

%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 561*x^5/5! +...

%e log(A(x)) = x + x^2/2! + 2*x^3/3! + 18*x^4/4! + 104*x^5/5! + 750*x^6/6! +...

%t CoefficientList[Series[1 + x - LambertW[-E^(x*(1+x))*x^3]/x,{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Feb 26 2014 *)

%o (PARI) {a(n,m=1)=n!*sum(k=0,n,sum(j=0,k,m*binomial(j+m,n-k)/(j+m)*binomial(n-k,k-j)*(k-j)^j/j!))}

%Y Cf. A162161.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 11 2009

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)