A162161 E.g.f. satisfies: A(x) = exp(x + x^2 + x^3*A(x)).

1, 1, 3, 13, 73, 561, 5251, 57583, 739089, 10794241, 176570371, 3209512791, 64116701353, 1396247370961, 32941566738627, 836962322583871, 22785381648804001, 661810614930630273, 20428823103775758595

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} (j+1)^(n-k-1)/(n-k)! * C(n-k,k-j)*C(k-j,j).

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

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).

E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A162697. [From Paul D. Hanna, Jul 18 2009]

E.g.f.: -LambertW(-exp(x*(1+x))*x^3)/x^3. - Vaclav Kotesovec, Jan 10 2014

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

Example

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

Mathematica

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

Prog

(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)))}

Crossrefs

Cf. A162697. [From Paul D. Hanna, Jul 18 2009]

Sequence in context: A205572 A128196 A367747 * A119013 A190878 A156154

Adjacent sequences: A162158 A162159 A162160 * A162162 A162163 A162164

Keyword

nonn

Author

Paul D. Hanna, Jun 26 2009

Status

approved