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 A007838 Number of permutations of n elements with distinct cycle lengths. 39
 1, 1, 1, 5, 14, 74, 474, 3114, 24240, 219456, 2231280, 23753520, 288099360, 3692907360, 51677246880, 775999798560, 12364465397760, 208583679951360, 3770392002048000, 71251563061002240, 1421847102467635200, 29861872557056870400, 655829140087057305600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 REFERENCES D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., Birkhäuser, Boston, 1982. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..450 (terms 0..200 from Vincenzo Librandi) Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006. A. Knopfmacher and R. Warlimont, Counting permutations and polynomials with a restricted factorization pattern, Australasian J. of Combinatorics, 13 (1996), 151-162. D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388. 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) FORMULA E.g.f.: Product_{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, Oct 13 2002 Asymptotics: a(n) ~ n!(e^{-g} + e^{-g}/n + O((log n)/n^2)), where g is the Euler gamma. E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018 MAPLE 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: # second Maple program: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1) +b(n-i, min(i-1, n-i))/i)) end: a:= n-> n!*b(n\$2): seq(a(n), n=0..23); # Alois P. Heinz, Feb 23 2022 MATHEMATICA max = 20; p = Product[(1 + x^m/m), {m, 1, max}]; s = Series[p, {x, 0, max}]; CoefficientList[s, x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011, after Maple *) PROG (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 */ CROSSREFS Cf. A000142, A080130, A087639, A088994, A317166. Sequence in context: A202764 A197876 A305017 * A306751 A305341 A316611 Adjacent sequences: A007835 A007836 A007837 * A007839 A007840 A007841 KEYWORD nonn AUTHOR EXTENSIONS More terms from James A. Sellers, Dec 24 1999 STATUS approved

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Last modified April 1 10:26 EDT 2023. Contains 361689 sequences. (Running on oeis4.)