A162161 - OEIS

%I #9 Feb 05 2025 00:36:47

%S 1,1,3,13,73,561,5251,57583,739089,10794241,176570371,3209512791,

%T 64116701353,1396247370961,32941566738627,836962322583871,

%U 22785381648804001,661810614930630273,20428823103775758595

%N E.g.f. satisfies: A(x) = exp(x + x^2 + x^3*A(x)).

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

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

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

%F E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A162697. - _Paul D. Hanna_, Jul 18 2009

%F E.g.f.: -LambertW(-exp(x*(1+x))*x^3)/x^3. - _Vaclav Kotesovec_, Jan 10 2014

%F a(n) ~ sqrt(2*r^2+r+3) * n^(n-1) / (exp(n) * r^(n+3)), where r = 0.542223654754281322169639... is the root of the equation exp(r^2+r+1)*r^3 = 1. - _Vaclav Kotesovec_, Jan 10 2014

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

%t CoefficientList[Series[-ProductLog[-E^(x*(1+x))*x^3]/x^3,{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 10 2014 *)

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

%Y Cf. A162697. - _Paul D. Hanna_, Jul 18 2009

%K nonn,changed

%O 0,3

%A _Paul D. Hanna_, Jun 26 2009