A115455 a(n) = number of reverse alternating fixed-point-free involutions w 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.

1, 0, 1, 1, 4, 13, 59, 308, 1871, 12879, 99144, 843735, 7865177, 79698760, 872235089, 10253148625, 128839087676, 1723418002261, 24450430660739, 366702601116524, 5796979684239647, 96339860422218143, 1679159568980521104, 30628034488033962287

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

R. P. Stanley, Permutations, Joint Mathematics Meeting, 2009.

R. P. Stanley, Alternating permutations and symmetric functions, J. Comb. Theory A 114 (3) (2007) 436-460

Formula

G.f.: (1-x^2)^{-1/4} (1+x)^{-1/2} Sum_{k>=0} E_{2k} v^k/k!, where E_{2k} is an Euler number and v = (1/4)*log((1+x)/(1-x)).

Example

a(3)=1 because there is one reverse alternating fixed-point-free involution on 1,...,6, viz., 351624.

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

Cf. A000085, A000111, A001147, A007779.

Sequence in context: A026663 A149486 A317891 * A241470 A219367 A149487

Adjacent sequences: A115452 A115453 A115454 * A115456 A115457 A115458

Keyword

easy,nonn

Author

Richard Stanley, Jan 22 2006

Status

approved