A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).

1, 1, 2, 4, 9, 22, 59, 172, 547, 1886, 7047, 28360, 122675, 567210, 2796999, 14641044, 81191947, 475148678, 2929442263, 18965690560, 128754649699, 914056305794, 6777666961735, 52367331911180, 421188392986843, 3519168714308702, 30519733808467031

Offset: 0

Author

Alois P. Heinz, Oct 26 2017

Keywords

• nonn

Examples

			a(3) = 4: 000, 001, 010, 011.
a(4) = 9: 0000, 0001, 0010, 0011, 0100, 0101, 0102, 0110, 0111.
a(5) = 22: 00000, 00001, 00010, 00011, 00100, 00101, 00102, 00110, 00111, 01000, 01001, 01002, 01010, 01011, 01020, 01021, 01022, 01100, 01101, 01102, 01110, 01111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..550

Crossrefs

Cf. A022493, A099960.

Programs

Maple

b:= proc(n, i, t, u) option remember; `if`(n<1, 1, add(
       b(n-1, j, t+`if`(j>i, 1, 0), `if`(i=j, u, 1-u)),
       j=`if`(u=0, i..t+1, 0..i)))
    end:
a:= n-> b(n-1, 0$3):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{j=0..n} binomial(n-1,j) * A099960(n-j).