A262745 Number of permutations of [n] with an odd number of rises.

0, 0, 1, 4, 12, 52, 360, 2656, 20160, 177472, 1814400, 20135296, 239500800, 3102326272, 43589145600, 654789062656, 10461394944000, 177738781376512, 3201186852864000, 60837094646972416, 1216451004088320000, 25542995336828157952, 562000363888803840000

Offset: 0

Author

Alois P. Heinz, Sep 29 2015

Keywords

• nonn

Examples

			a(2) = 1: 12.
a(3) = 4: 132, 213, 231, 312.
		

Links

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

Crossrefs

Cf. A000142, A128103.

Programs

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$2):
seq(a(n), n=0..25);

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, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 01 2021, after Alois P. Heinz *)

Formula

E.g.f.: (1/(1-x) -1 -tanh(x))/2.

a(n) = A000142(n) - A128103(n).