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A161631 E.g.f. satisfies: A(x) = 1 + x*exp(x*A(x)). 1
1, 1, 2, 9, 52, 425, 4206, 50827, 713000, 11500785, 208833850, 4226139731, 94226705772, 2296472176297, 60727113115046, 1732020500240955, 52998549321251536, 1731977581804704737, 60205422811336194546 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.

E.g.f.: A(x) = 1 + Sum_{n>=1} x^(2*n-1) * n^(n-1) * exp(n*x) / n!.

E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.

a(n) = n*A125500(n-1) for n>0, where exp(x*A(x)) = e.g.f. of A125500.

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

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

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

a(n) ~ sqrt(1+LambertW(1/(2*sqrt(exp(1))))) * n^(n-1) / (exp(n) * 2^(n+1/2) * (LambertW(1/(2*sqrt(exp(1)))))^(n+1)). - Vaclav Kotesovec, Jul 09 2013

EXAMPLE

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! +...

exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +...

MATHEMATICA

CoefficientList[Series[1-LambertW[-x^2*E^x]/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

PROG

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

CROSSREFS

Cf. A125500.

Sequence in context: A143922 A305304 A110322 * A121678 A124347 A266469

Adjacent sequences:  A161628 A161629 A161630 * A161632 A161633 A161634

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 18 2009

STATUS

approved

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Last modified September 21 09:44 EDT 2020. Contains 337268 sequences. (Running on oeis4.)