A346194 Total sum of the left-to-right strict peak maxima in all Dyck paths of semilength n.

0, 1, 3, 11, 40, 148, 555, 2100, 7997, 30605, 117602, 453421, 1753176, 6795248, 26393431, 102702230, 400277998, 1562292741, 6105426033, 23887275883, 93554945414, 366754396228, 1438986625349, 5650409534767, 22203298031827, 87306238753663, 343511939707274

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+2) + 2 + 2 + 3 = 11:
                                         /\
                  /\    /\      /\/\    /  \
       /\/\/\  /\/  \  /  \/\  /    \  /    \ .

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

Crossrefs

Cf. A000108, A000302, A008549, A346157, A346195.

Sequence in context: A296221 A371872 A014301 * A346317 A119375 A149063

Adjacent sequences: A346191 A346192 A346193 * A346195 A346196 A346197

Keyword

nonn

Author

Alois P. Heinz, Jul 09 2021

Status

approved