A287860 Number of Dyck paths of semilength 2n such that the maximal number of peaks per level equals n.

1, 1, 7, 29, 163, 925, 5580, 34751, 222627, 1456952, 9699872, 65474460, 446971110, 3080074508, 21393773841, 149614083615, 1052537452164, 7443584137525, 52888757972865, 377382278671610, 2703141489113003, 19430405608302831, 140118758417377105

Offset: 0

Author

Alois P. Heinz, Jun 01 2017

Keywords

• nonn

Examples

			.                  /\       /\         /\/\
.  a(2) = 7:  /\/\/  \   /\/  \/\   /\/    \
.
.                                     /\/\
.   /\         /\  /\     /\/\       /    \
.  /  \/\/\   /  \/  \   /    \/\   /      \  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A287822.

Programs

Maple

b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
      b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
       m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
    end:
g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
a:= n-> `if`(n=0, 1, g(2*n, n)-g(2*n, n-1)):
seq(a(n), n=0..23);

Mathematica

b[n_, k_, j_] := b[n, k, j] = If[j == n, 1, Sum[b[n - j, k, i]*Sum[ Binomial[i, m] * Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, 1, Min[j + k, n - j]}]];
g[n_, k_] := g[n, k] = Sum[b[n, k, j], {j, 1, k}];
a[n_] := If[n == 0, 1, g[2*n, n] - g[2*n, n - 1]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A287822(2n,n).