OFFSET
0,4
COMMENTS
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) != i for all i. - Richard Stanley, 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). - Richard Stanley, Jan 22 2006
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 545.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
W. F. Galway, 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, arXiv:math/0603520 [math.CO], 2006.
R. P. Stanley, Permutations
FORMULA
Sum_{n>=0} a(n)*x^n = (1-x^2)^(-1/4)*sqrt(1+x)*Sum_{k>=0} E_{2k} v^k/k!, where E_{2k} is an Euler number and v = (1/4)*log((1+x)/(1-x)). - Richard Stanley, Jan 22 2006
Berndt gives an explicit g.f. on page 547.
MATHEMATICA
Table[SeriesCoefficient[(1-x^2)^(-1/4)*(1+x)^(1/2)*Sum[(-1)^k*EulerE[2*k]*(1/4*Log[(1+x)/(1-x)])^k/k!, {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 29 2014 *)
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
William F. Galway (galway(AT)math.uiuc.edu)
EXTENSIONS
Edited by Ralf Stephan, May 08 2007
STATUS
approved