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

0,4

Links

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

Formula

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

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

Example

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

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 *)

Crossrefs

Cf. A000142, A128103.

Sequence in context: A065525 A317281 A190342 * A180381 A149413 A149414

Adjacent sequences: A262742 A262743 A262744 * A262746 A262747 A262748

Keyword

nonn

Author

Alois P. Heinz, Sep 29 2015

Status

approved