A138013 E.g.f. satisfies: A(x) = 1 - log(1 - x*A(x)).

1, 1, 3, 17, 146, 1694, 24834, 440586, 9180800, 219829536, 5948287560, 179508872520, 5978006444112, 217772950035120, 8614798644364080, 367768502385434640, 16852524904388586240, 825075552824125305600, 42981992589364756939008, 2373967488394457834095872

Offset

0,3

Comments

a(n) = A038037(n+1)/(n+1) for n>=0 where A038037(n) is the number of labeled rooted compound windmills (mobiles) with n nodes.

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: A(x) = (1/x)*Series_Reversion[ x/(1 - log(1-x)) ].

E.g.f.: A(x) = 1 + Series_Reversion( (1-exp(-x))/(1+x) ).

E.g.f. A(x) satisfies: exp(1 - A(x)) = 1 - x*A(x).

a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^n * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013

a(n) = sum(n!/(n+1-k)! * |stirling1(n,k)|, k=0..n). - Michael D. Weiner, Dec 23 2014

Example

E.g.f.: A(x) = 1 + x + 3x^2/2! + 17x^3/3! + 146x^4/4! + 1694x^5/5! + ...
where A(x) = 1 - log(1 - x*A(x)):
A(x) = 1 + x*A(x) + x^2*A(x)^2/2 + x^3*A(x)^3/3 +...+ x^n*A(x)^n/n +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1-E^(-x))/(1+x), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(1/x*serreverse(x/(1-log(1-x + x*O(x^n) ))), n+1)}
(PARI) {a(n)=n!*polcoeff(1 + serreverse((1-exp(-x+x^2*O(x^n)))/(1+x +x*O(x^n))), n)}

Crossrefs

Cf. A038037.

Sequence in context: A241805 A368233 A277466 * A052807 A080253 A234289

Adjacent sequences: A138010 A138011 A138012 * A138014 A138015 A138016

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2008

Status

approved