A242164 Number of ascent sequences of length 2n with exactly n flat steps.

1, 1, 3, 20, 175, 2016, 28182, 465036, 8828820, 189496450, 4535113154, 119706872376, 3454013050488, 108140144894600, 3650830138093500, 132194177662402800, 5110163818369981650, 210037720563156731850, 9146299175093615073000, 420627290039763259876500

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, May 05 2014

Keywords

• nonn

Examples

			a(0) = 1: the empty sequence.
a(1) = 1: [0,0].
a(2) = 3: [0,0,0,1], [0,0,1,1], [0,1,1,1].
a(3) = 20: [0,0,0,0,1,0], [0,0,0,0,1,2], [0,0,0,1,0,0], [0,0,0,1,1,0], [0,0,0,1,1,2], [0,0,0,1,2,2], [0,0,1,0,0,0], [0,0,1,1,0,0], [0,0,1,1,1,0], [0,0,1,1,1,2], [0,0,1,1,2,2], [0,0,1,2,2,2], [0,1,0,0,0,0], [0,1,1,0,0,0], [0,1,1,1,0,0], [0,1,1,1,1,0], [0,1,1,1,1,2], [0,1,1,1,2,2], [0,1,1,2,2,2], [0,1,2,2,2,2].
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..70

Crossrefs

Cf. A242153.

Programs

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[ If[j == i, x, 1]*b[n - 1, j, t + If[j > i, 1, 0]], {j, 0, t + 1}]]]; a[n_] := Coefficient[b[2n, -1, -1], x, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

Formula

a(n) = A242153(2n,n).

a(n) ~ 6*sqrt(3) / (Pi^3 * exp(Pi^2/12)) * (24/Pi^2)^n * n!. - Vaclav Kotesovec, Aug 28 2014