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A346195
Total sum of the left-to-right weak peak maxima in all Dyck paths of semilength n.
3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 09 2021
STATUS
approved