A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

1, 1, -2, 9, -68, 740, -10554, 185906, -3891320, 94259952, -2592071760, 79748398752, -2713685928744, 101184283477680, -4102325527316184, 179674073609647080, -8454031849605513024, 425281651659459346944, -22777115050468598701248

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..375

Formula

E.g.f.: A(x) = log(1+x)/LambertW(log(1+x)).

log(A(x)) = LambertW(log(1+x)).

E.g.f.: A(x) = 1/G(-x) where G(x) = g.f. of A052813.

E.g.f. of A052807 = -log(A(-x)) = -log(1-x)/A(-x).

a(n) = Sum_{k=0..n} (-1)^(k+1)*Stirling1(n,k)*(k-1)^(k-1). - Vladeta Jovovic, Jul 22 2006

|a(n)| ~ exp((exp(-1)-1)*n+3/2) * n^(n-1) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

Mathematica

CoefficientList[Series[Log[1+x]/LambertW[Log[1+x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Table[StirlingS1[n, 0] + StirlingS1[n, 1] + Sum[(-1)^(k + 1)*StirlingS1[n, k]*(k - 1)^(k - 1), {k, 2, n}], {n, 0, 50}] (* G. C. Greubel, Jun 21 2017 *)
CoefficientList[Series[Exp[LambertW[Log[1+x]]], {x, 0, 25}], x]* Range[0, 25]! (* G. C. Greubel, Jun 22 2017 *)

Prog

(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[ #A]=-Vec(Ser(A)^Ser(A))[ #A]); n!*A[n+1]}
(PARI) x='x+O('x^50); Vec(serlaplace(exp(lambertw(log(1+x))))) \\ G. C. Greubel, Jun 22 2017

Crossrefs

Cf. A008275, A052813, A052807, A349561, A349583, A349585.

Sequence in context: A038037 A138212 A134261 * A020563 A354730 A193160

Adjacent sequences: A120977 A120978 A120979 * A120981 A120982 A120983

Keyword

sign

Author

Paul D. Hanna, Jul 20 2006

Status

approved