A287989 Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and the number of peaks of adjacent levels is different.

1, 1, 1, 1, 6, 10, 27, 84, 226, 770, 2390, 7579, 25222, 84299, 285284, 976105, 3386494, 11858759, 41782516, 148205047, 529101609, 1899680494, 6854597493, 24847293152, 90460431604, 330654288724, 1213033321450, 4465027739962, 16486012746085, 61044028354833

Offset: 0

Author

Alois P. Heinz, Jun 04 2017

Keywords

• nonn

Examples

			. (4) = 6:
.                    /\      /\      /\          /\/\    /\/\
.     /\/\/\/\  /\/\/  \  /\/  \/\  /  \/\/\  /\/    \  /    \/\  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A000108, A287993.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
       b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
       t={$max(1, i-j)..min(n-j, i-1)} minus {k}), i=1..n-j))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
seq(a(n), n=0..30);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ {k}}], {i, 1, n - j}]];
a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from Maple *)