0,4

Also a(n) = number of alternating fixed-point-free involutions on 1,2,...,2n, i.e. w(1)>w(2)<w(3)>w(4)<...>w(2n), w^2=1 and w(i) not= i for all i. - R. P. Stanley (rstan(AT)math.mit.edu), Jan 22 2006. For example, a(3)=2 because there are two alternating fixed-point-free involutions on 1,...,6, viz., 214365 and 645231.

If b(n) is the number of reverse alternating fixed-point-free involutions on 1,2,...,2n (A115455) then b(n-1)+b(n)=a(n). - R. P. Stanley (rstan(AT)math.mit.edu), Jan 22 2006

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 545.

Galway, W. F., An Asymptotic Expansion of Ramanujan, in Number Theory (Fifth Conference of Canadian Number Theory Assoc., August, 1996, Carleton University), pp. 107-110, ed. R. Gupta and K. S. Williams, Amer. Math. Soc., 1999.

R. P. Stanley, Alternating permutations and symmetric functions

Sum_{n=0..infinity} a(n)x^n = (1-x^2)^{-1/4} (1+x)^{1/2} sum_{k=0..infinity) E_{2k} v^k/k!, where E_{2k} is an Euler number and v = (1/4)log((1+x)/(1-x)) - R. P. Stanley (rstan(AT)math.mit.edu), Jan 22 2006

Berndt gives an explicit g.f. on page 547.

Cf. A115455.

Sequence in context: A005967 A104859 A108289 this_sequence A084161 A102038 A002135

Adjacent sequences: A007776 A007777 A007778 this_sequence A007780 A007781 A007782

nonn,nice,easy

William F. Galway [ galway(AT)math.uiuc.edu ]

Edited by Ralf Stephan, May 08 2007