A262167 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= 5.

1, 1, 2, 5, 20, 87, 521, 3264, 25899, 212534, 2092218, 21250451, 249294149, 3018733862, 41077515364, 577524896681, 8940290166542, 143098583946093, 2483312451690110, 44571301924473611, 857112705946351481, 17044616630699383294, 359813788663496645489

Offset: 0

Author

Alois P. Heinz, Sep 13 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..453

Crossrefs

Column k=5 of A262163.

Programs

Maple

b:= proc(u, o, c) option remember; `if`(c<0 or c>5, 0, `if`(u+o=0,
       x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..5))(add(
       b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..min(n, 5)))(b(0, n, 0)):
seq(a(n), n=0..25);

Mathematica

b[u_, o_, c_] := b[u, o, c] = If[c < 0 || c > 5, 0, If[u + o == 0, x^c, Function[p, Sum[Coefficient[p, x, i]*x^Max[i, c], {i, 0, 5}]][Sum[b[u - j, o - 1 + j, c - 1], {j, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, o}]]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, 5]}]][b[0, n, 0]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Formula

a(n) = A262163(n,5).