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

1, 1, 2, 6, 36, 260, 2190, 23562, 294056, 4145976, 66518010, 1187366510, 23307288972, 500683995396, 11669239646246, 293211947901810, 7905976017270480, 227653751742812912, 6972326784534024306

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

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!.

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

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!.

...

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

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

E.g.f.: 1 + x - LambertW(-exp(x*(1+x))*x^3)/x. - Vaclav Kotesovec, Feb 26 2014

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

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 561*x^5/5! +...
log(A(x)) = x + x^2/2! + 2*x^3/3! + 18*x^4/4! + 104*x^5/5! + 750*x^6/6! +...

Mathematica

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

Prog

(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!))}

Crossrefs

Cf. A162161.

Sequence in context: A351825 A358080 A369091 * A107099 A143021 A007657

Adjacent sequences: A162694 A162695 A162696 * A162698 A162699 A162700

Keyword

nonn

Author

Paul D. Hanna, Jul 11 2009

Status

approved