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%I #23 Jun 04 2022 07:13:26
%S 0,1,4,15,57,218,837,3224,12455,48244,187307,728692,2839877,11084756,
%T 43325744,169548783,664229072,2604770882,10223744018,40161025704,
%U 157878855072,621070768564,2444741008686,9628942865104,37945470536353,149609922922904,590153796979867
%N Total sum of the left-to-right weak 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="/A346195/b346195.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+1) + (1+2) + 2 + (2+2) + 3 = 15:
%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 Function[p, p + {0, If[t == 1, p[[1]]*h, 0]}][b[x-1, y-1, 0, h]], 0] +
%t If[y < x-1, b[x-1, y+1, If[y+1 >= h, 1, 0], Max[h, y+1]], 0]];
%t a[n_] := b[2*n, 0, 0, 0][[2]];
%t Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Jun 04 2022, after _Alois P. Heinz_ *)
%Y Cf. A000108, A000302, A008549, A346158, A346194.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 09 2021
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