A346194 - OEIS

%I #21 Apr 04 2022 16:49:50

%S 0,1,3,11,40,148,555,2100,7997,30605,117602,453421,1753176,6795248,

%T 26393431,102702230,400277998,1562292741,6105426033,23887275883,

%U 93554945414,366754396228,1438986625349,5650409534767,22203298031827,87306238753663,343511939707274

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

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

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

%H Alois P. Heinz, <a href="/A346194/b346194.txt">Table of n, a(n) for n = 0..650</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%e a(3) = 1 + (1+2) + 2 + 2 + 3 = 11:

%e                                          /\

%e                   /\    /\      /\/\    /  \

%e        /\/\/\  /\/  \  /  \/\  /    \  /    \ .

%p b:= proc(x, y, t, h) option remember; `if`(x=0, [1, 0], `if`(y>0,

%p      (p-> p+[0, `if`(t=1, p[1]*h, 0)])(b(x-1, y-1, 0, h)), 0)+

%p      `if`(y<x-1, b(x-1, y+1, `if`(y+1>h, 1, 0), max(h, y+1)), 0))

%p     end:

%p a:= n-> b(2*n, 0$3)[2]:

%p seq(a(n), n=0..32);

%t b[x_, y_, t_, h_] := b[x, y, t, h] = If[x == 0, {1, 0}, If[y > 0,

%t    With[{p = b[x-1, y-1, 0, h]}, p+{0, If[t == 1, p[[1]]*h, 0]}]], {0, 0}]+

%t    If[y < x - 1, b[x-1, y+1, If[y+1 > h, 1, 0], Max[h, y+1]], {0, 0}] /.

%t    Null -> 0;

%t a[n_] := b[2*n, 0, 0, 0][[2]];

%t Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Apr 04 2022, after _Alois P. Heinz_ *)

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

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 09 2021