OFFSET
0,3
COMMENTS
The function sqrt(-log(1-q)/q) is associated with the generating function of connected graphs enumeration
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
P. Flajolet, B. Salvy, and G. Schaeffer, Airy Phenomena and Analytic Combinatorics of Connected Graphs, Electronic Journal of Combinatorics, Volume 11(1), 2004, Research Paper #R34 page 8.
FORMULA
a(n) = numerator(Sum_{k=0..n} binomial(k+n-1/2,k)*binomial(-n-k-3/2,n-k)*Stirling2(n+k,k)*k!/((2*n+1)*(n+k)!)). - Tani Akinari, Oct 23 2024
EXAMPLE
Coefficients begin: 1, 1/4, 13/96, 35/384, 6271/92160, ...
MAPLE
gf := sqrt(-log(1-q)/q) ;
taylor(%, q=0, 18) ;
gfun[seriestolist](%) ;
map(numer, %) ; # R. J. Mathar, Mar 15 2018
MATHEMATICA
s = Sqrt[Log[1/(1 - q)]/q] + O[q]^20; CoefficientList[s, q] // Numerator
PROG
(PARI) Vec(apply(numerator, sqrt(-log(1-'q)/'q))) \\ M. F. Hasler, Mar 16 2018
(PARI) a(n)=numerator(sum(k=0, n, binomial(k+n-1/2, k)*binomial(-n-k-3/2, n-k)*stirling(n+k, k, 2)*k!/((2*n+1)*(n+k)!))) \\ Tani Akinari, Oct 23 2024
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Jean-François Alcover, Apr 20 2016
EXTENSIONS
Definition corrected by Juan Arias-de-Reyna, Mar 15 2018
Edited by M. F. Hasler, Mar 16 2018
STATUS
approved