0,4

Philippe Flajolet, Eric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math.CO/0606370

D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., Birkhaeuser, Boston, 1982.

A. Knopfmacher and R. Warlimont, Counting permutations and polynomials with a restricted factorization pattern, Australasian J. of Combinatorics, 13 (1996), 151-162.

A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Examples 8.10 and 11.8 (pdf, ps)

P. Flajolet et al., A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics

E.g.f.: prod {m >= 1} (1+x^m/m).

a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)^(1-k/d) and a(0) = 1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 13 2002

Asymptotics: a(n) ~ n!(e^{-g} + e^{-g}/n + O((log n)/n^2)), where g is the Euler gamma

a(n) ~ exp(3/2 cbrt(2 pi n))/sqrt(n).

p := product((1+x^m/m), m=1..100): s := series(p, x, 100): for i from 1 to 100 do printf(`%.0f, `, i!*coeff(s, x, i)) od:

(PARI) {a(n)=if(n<0, 0, n!*polcoeff( prod(k=1, n, 1+x^k/k, 1+x*O(x^n)), n))} /* Michael Somos Sep 19 2006 */

Adjacent sequences: A007835 A007836 A007837 this_sequence A007839 A007840 A007841

Sequence in context: A127922 A004030 A128102 this_sequence A024167 A077262 A058072

nonn

Arnold Knopfmacher [ ARNOLDK(AT)gauss.cam.wits.ac.za ]

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999