A250260 The number of 5-alternating permutations of [n].

1, 1, 1, 2, 3, 4, 5, 29, 133, 412, 1041, 2300, 22991, 170832, 822198, 3114489, 10006375, 141705439, 1457872978, 9522474417, 48094772656, 202808749375, 3716808948931, 48860589990687, 403131250565618, 2545098156762649, 13287626090593750

Offset: 0

Author

R. J. Mathar, Nov 15 2014

Keywords

• nonn

Comments

A sequence a(1), a(2),... is called k-alternating if a(i) > a(i+1) iff i=1 (mod k).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, page 17.

Crossrefs

Cf. A065619 (2-alternating), A249402 (3-alternating), A250259 (4-alternating).

Column k=5 of A250261.

Programs

Maple

# dowupP defined in A250259.
A250260 :=proc(n)
    downupP(n,4) ;
end proc:
seq(A250260(n),n=0..20) ;
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=1, add(b(u-j, o+j-1, irem(t+1, 5)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 5)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 15 2014

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, If[t == 1, Sum[b[u-j, o+j-1, Mod[t+1, 5]], {j, 1, u}], Sum[b[u+j-1, o-j, Mod[t+1, 5]], {j, 1, o}]]]; a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)