A128103 Number of permutations of [n] with an even number of rises.

1, 1, 1, 2, 12, 68, 360, 2384, 20160, 185408, 1814400, 19781504, 239500800, 3124694528, 43589145600, 652885305344, 10461394944000, 177948646719488, 3201186852864000, 60808005761859584, 1216451004088320000, 25547946834881282048, 562000363888803840000

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

F. C. S. Brown, T. M. A. Fink and K. Willbrand, On arithmetic and asymptotic properties of up-down numbers, arXiv:math/0607763 [math.CO], 2006.

Formula

E.g.f.: 1 + 1/2 [z/(1-z) + tanh(z) ].

a(n) = A000142(n) - A262745(n).

If n is even, a(n) = (n)!/2 (A002674), if n is odd, a(n) = (n)! * (1 + (-1)^((n-1)/2) * A002430((n+1)/2) / A036279((n+1)/2)) / 2. - Michel Marcus, Dec 09 2012

Conjecture: a(n) = Sum_{k = 0..n} Sum_{j = 0..k} (-1)^(n+j)*binomial(n,k-j)*j^n. - Peter Bala, Jan 22 2020

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, t,
      add(b(u-j, o+j-1, t), j=1..u)+
      add(b(u+j-1, o-j, 1-t), j=1..o))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 29 2015

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[u - j, o + j - 1, t], {j, 1, u}] + Sum[b[u + j - 1, o - j, 1 - t], {j, 1, o}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *)

Prog

(PARI) x='x+O('x^99); Vec(serlaplace((x/(1-x)+tanh(x))/2+1)) \\ Altug Alkan, Jul 25 2017

Crossrefs

Cf. A000142, A002674, A002430, A036279, A262745.

Sequence in context: A056636 A174327 A245997 * A359489 A329789 A245854

Adjacent sequences: A128100 A128101 A128102 * A128104 A128105 A128106

Keyword

nonn

Author

Ralf Stephan, May 09 2007

Extensions

More terms from Alois P. Heinz, Sep 29 2015

Status

approved