0,3

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

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

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.

B. A. Kuperschmidt, ... And free lunch for all.

B. A. Kuperschmidt, Journal of Non linear Mathematical Physics 2000 v.7 no.2, A Review of Bruce C.Berndt's Ramanujan's Notebooks parts I-V

E.g.f.: log((1+x)/(1-x))/(2(1-x)). a(n) = n! sum[ k=0..n, k odd ] 1/k.

a(n) = n!/2*(Psi(ceil(n/2)+1/2)+gamma+2*ln(2)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 20 2003

a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*2^(k-1)*binomial(n, k)/k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 12 2005

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

(PARI) {a(n)=if(n<0, 0, n!*sum(k=1, n, (k%2)/k))} /* Michael Somos Sep 19 2006 */

Cf. A049034(n)=a(2n+1).

Sequence in context: A081561 A009753 A141202 this_sequence A048855 A062797 A134751

Adjacent sequences: A081355 A081356 A081357 this_sequence A081359 A081360 A081361

nonn

Michael Somos, Mar 18 2003