A081358 - OEIS

%I #44 Mar 17 2018 04:13:20

%S 0,1,2,8,32,184,1104,8448,67584,648576,6485760,74972160,899665920,

%T 12174658560,170445219840,2643856588800,42301705420800,

%U 740051782041600,13320932076748800,259500083163955200,5190001663279104000,111422936937037824000,2451304612614832128000

%N E.g.f.: log((1+x) / (1-x)) / (2*(1-x)).

%C 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_, Aug 12 2004

%C a(n) is a function of the harmonic numbers. a(n) = n!*h(n) - n!/2 * h(floor(n/2)), where h(n) = Sum_{k=1..n} 1/k. - _Gary Detlefs_, Aug 06 2010

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 3.3.13

%H N. J. A. Sloane, <a href="/A081358/b081358.txt">Table of n, a(n) for n = 0..30</a>

%H B. A. Kuperschmidt, <a href="http://www.sm.luth.se/~norbert/home_journal/electronic/72brev3.ps">... And free lunch for all.</a>

%H B. A. Kuperschmidt, Journal of Nonlinear Mathematical Physics 2000 v. 7 no. 2, <a href="http://arXiv.org/abs/math.HO/0004188">A Review of Bruce C. Berndt's Ramanujan's Notebooks parts I-V</a>

%F E.g.f.: log((1+x) / (1-x)) / (2*(1-x)).

%F a(n) = n! * Sum_{k=0..n, k odd} 1/k.

%F a(n) = n!/2*(Psi(ceiling(n/2)+1/2)+gamma+2*log(2)). - _Vladeta Jovovic_, Oct 20 2003

%F a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*2^(k-1)*binomial(n, k)/k. - _Vladeta Jovovic_, Aug 12 2005

%F a(n) = n*a(n-1) + ((-1)^(n+1)+1)/2*(n-1)!. - _Gary Detlefs_, Aug 06 2010

%F a(n) = A000254(n) - A092691(n). - _Gary Detlefs_, Aug 06 2010

%F a(n) ~ n!/2 * (log(n) + gamma + log(2)), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 05 2013

%F a(2*n + 1) = A049034(n).

%F E.g.f.: arctanh(x)/(1 - x). - _Ilya Gutkovskiy_, Dec 19 2017

%e G.f. = x + 2*x^2 + 8*x^3 + 32*x^4 + 184*x^5 + 1104*x^6 + 8448*x^7 + ...

%t nn = 20; Range[0, nn]! CoefficientList[

%t   D[Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], y] /. y -> 1, x]  (* _Geoffrey Critzer_, Aug 27 2012 *)

%t a[ n_] := If[ n < 0, 0, n! Sum[ 1/k, {k, 1, n, 2}]]; (* _Michael Somos_, Jan 06 2015 *)

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ (1 + x) / (1 - x)] / (2 (1 - x)), {x, 0, n}]]; (* _Michael Somos_, Jan 06 2015 *)

%o (PARI) {a(n) = if( n<1, 0, n! * polcoeff( log(1 + 2 / (-1 + 1 / (x + x * O(x^n)))) / (2 * (1-x)), n))};

%o (PARI) {a(n) = if( n<0, 0, n! * sum(k=1, n, (k%2)/k))}; /* _Michael Somos_, Sep 19 2006 */

%o (PARI) first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)/(1 - x)), -n) \\ _Iain Fox_, Dec 19 2017

%Y Cf. A000254, A001640, A049034, A092691, A151884.

%K nonn

%O 0,3

%A _Michael Somos_, Mar 18 2003