A052813 Expansion of e.g.f.: LambertW(log(1-x))/log(1-x).

1, 1, 4, 27, 260, 3280, 51414, 965762, 21175496, 531317520, 15021531840, 472654558992, 16385500397496, 620612495460048, 25500923655523848, 1129909190812470840, 53705490284841870144, 2725878142900911376896

Offset

0,3

Comments

Previous name was: A simple grammar.

Given e.g.f. A(x), log(A(x)) = -log(1-x)*A(x) equals e.g.f. of A052807. - Paul D. Hanna, Jul 19 2006

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 778

Formula

E.g.f.: -1/log(-1/(-1+x))*LambertW(-log(-1/(-1+x))).

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003

E.g.f. A(x) satisfies: A(x) = 1/(1-x)^A(x). - Paul D. Hanna, Jul 19 2006

E.g.f.: Sum_{n>=0} (n+1)^(n-1)*(-log(1-x))^n/n!. - Paul D. Hanna, Jun 22 2009

E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)|*A(x)^k. - Paul D. Hanna, Jan 16 2013

a(n) ~ n^(n-1) * exp(3/2+n*exp(-1)-n) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Sep 30 2013

E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x) + k). - Paul D. Hanna, Oct 26 2015

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 260*x^4/4! +...
Log(A(x))/A(x) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...

Maple

spec := [S, {C=Cycle(Z), S=Set(B), B=Prod(C, S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

CoefficientList[Series[1/Log[1-x]*LambertW[Log[1-x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/(1-x+x*O(x^n))^A); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 19 2006
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, (m+1)^(m-1)/m!*(-log(1-x+x*O(x^n)))^m), n)} \\ Paul D. Hanna, Jun 22 2009
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 26 2015
(PARI) x='x+O('x^30); Vec(serlaplace((1/log(1-x))*lambertw(log(1-x)))) \\ G. C. Greubel, Feb 19 2018

Crossrefs

Cf. A052807 (log(A(x)).

Sequence in context: A301335 A362701 A177379 * A218653 A359461 A121353

Adjacent sequences: A052810 A052811 A052812 * A052814 A052815 A052816

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

New name using e.g.f. from Vaclav Kotesovec, Sep 30 2013

Status

approved