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A272108
Numerators of coefficients of the series of sqrt(-log(1-q)/q).
2
1, 1, 13, 35, 6271, 2211, 2760011, 1875133, 557576779, 13761972821, 3244313727791, 185892006151, 77616670784995799, 3796517116479937, 1261429981603803133, 4682424392750614127, 71254131149637283370867, 385105979676360971447, 2401927688678818378270288001
OFFSET
0,3
COMMENTS
The function sqrt(-log(1-q)/q) is associated with the generating function of connected graphs enumeration
LINKS
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
Cf. A272109 (denominators).
Sequence in context: A221592 A135172 A183309 * A034119 A054285 A101103
KEYWORD
nonn,frac
AUTHOR
EXTENSIONS
Definition corrected by Juan Arias-de-Reyna, Mar 15 2018
Edited by M. F. Hasler, Mar 16 2018
STATUS
approved