Search: id:A081358
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%I A081358
%S A081358 0,1,2,8,32,184,1104,8448,67584,648576,6485760,74972160,899665920,
%T A081358 12174658560,170445219840,2643856588800,42301705420800,740051782041600,
%U A081358 13320932076748800,259500083163955200,5190001663279104000
%N A081358 Expansion of log((1+x)/(1-x))/(2(1-x)).
%C A081358 Number of cycles of odd cardinality in all permutations of [n]. Example: 
               a(3)=8 because among (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), 
               (123) we have eight cycles of odd length. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 12 2004
%D A081358 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 
               1983, Exercise 3.3.13
%D A081358 B. A. Kuperschmidt, ... And free lunch for all. A review of Bruce C. 
               Berndt's Ramanujan's notebooks, J. Nonlinear Math. Phys., 7 (2000), 
               R7-R37. MR 2002d:33024.
%H A081358 N. J. A. Sloane, <a href="b081358.txt">Table of n, a(n) for n = 0..30</
               a>
%H A081358 B. A. Kuperschmidt, <a href="http://www.sm.luth.se/~norbert/home_journal/
               electronic/72brev3.ps">... And free lunch for all.</a>
%H A081358 B. A. Kuperschmidt, Journal of Non linear Mathematical Physics 2000 v.7 
               no.2, <a href="http://arXiv.org/pdf/math.HO/0004188">A Review of 
               Bruce C.Berndt's Ramanujan's Notebooks parts I-V</a>
%F A081358 E.g.f.: log((1+x)/(1-x))/(2(1-x)). a(n) = n! sum[ k=0..n, k odd ] 1/k.
%F A081358 a(n) = n!/2*(Psi(ceil(n/2)+1/2)+gamma+2*ln(2)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 20 2003
%F A081358 a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*2^(k-1)*binomial(n, k)/k. - Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Aug 12 2005
%o A081358 (PARI) a(n)=if(n<1,0,n!*polcoeff(log(1+2/(-1+1/(x+x*O(x^n))))/(1-x)/2,
               n))
%o A081358 (PARI) {a(n)=if(n<0, 0, n!*sum(k=1, n, (k%2)/k))} /* Michael Somos Sep 
               19 2006 */
%Y A081358 A049034(n)=a(2n+1). Cf. A151884, A092691.
%Y A081358 Sequence in context: A081561 A009753 A141202 this_sequence A048855 A062797 
               A134751
%Y A081358 Adjacent sequences: A081355 A081356 A081357 this_sequence A081359 A081360 
               A081361
%K A081358 nonn
%O A081358 0,3
%A A081358 Michael Somos, Mar 18 2003

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