A346195 Total sum of the left-to-right weak peak maxima in all Dyck paths of semilength n.

0, 1, 4, 15, 57, 218, 837, 3224, 12455, 48244, 187307, 728692, 2839877, 11084756, 43325744, 169548783, 664229072, 2604770882, 10223744018, 40161025704, 157878855072, 621070768564, 2444741008686, 9628942865104, 37945470536353, 149609922922904, 590153796979867

Offset

0,3

Comments

Sum of all peak heights in Dyck paths of semilength n is A000302(n-1) for n>0.

Sum of all heights in Dyck paths of semilength n is A008549(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..650

Wikipedia, Counting lattice paths

Example

a(3) = (1+1+1) + (1+2) + 2 + (2+2) + 3 = 15:
                                         /\
                  /\    /\      /\/\    /  \
       /\/\/\  /\/  \  /  \/\  /    \  /    \ .

Maple

b:= proc(x, y, t, h) option remember; `if`(x=0, [1, 0], `if`(y>0,
     (p-> p+[0, `if`(t=1, p[1]*h, 0)])(b(x-1, y-1, 0, h)), 0)+
     `if`(y<x-1, b(x-1, y+1, `if`(y+1>=h, 1, 0), max(h, y+1)), 0))
    end:
a:= n-> b(2*n, 0$3)[2]:
seq(a(n), n=0..32);

Mathematica

b[x_, y_, t_, h_] := b[x, y, t, h] = If[x == 0, {1, 0}, If[y > 0,
     Function[p, p + {0, If[t == 1, p[[1]]*h, 0]}][b[x-1, y-1, 0, h]], 0] +
     If[y < x-1, b[x-1, y+1, If[y+1 >= h, 1, 0], Max[h, y+1]], 0]];
a[n_] := b[2*n, 0, 0, 0][[2]];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 04 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000108, A000302, A008549, A346158, A346194.

Sequence in context: A047108 A125145 A242781 * A371854 A277924 A371777

Adjacent sequences: A346192 A346193 A346194 * A346196 A346197 A346198

Keyword

nonn

Author

Alois P. Heinz, Jul 09 2021

Status

approved