cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, 1, 4, 27, 260, 3280, 51414, 965762, 21175496, 531317520, 15021531840, 472654558992, 16385500397496, 620612495460048, 25500923655523848, 1129909190812470840, 53705490284841870144, 2725878142900911376896
Offset: 0

Views

Author

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

Keywords

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

Examples

			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 +...
		

Crossrefs

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

Programs

  • 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 *)
  • 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

Formula

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

Extensions

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