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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Wikipedia, Counting lattice paths

Formula

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

Example

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

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 *)

Crossrefs

Cf. A287822.

Sequence in context: A179599 A266473 A297677 * A347814 A175208 A307954

Adjacent sequences: A287857 A287858 A287859 * A287861 A287862 A287863

Keyword

nonn

Author

Alois P. Heinz, Jun 01 2017

Status

approved