A371903 Total number of levels in all Dyck paths of semilength n containing exactly 2 path nodes.

0, 1, 3, 5, 15, 44, 134, 427, 1408, 4753, 16321, 56812, 200046, 711425, 2551886, 9222147, 33544682, 122712465, 451169747, 1666248405, 6178586630, 22994275870, 85859249486, 321562877934, 1207665205311, 4547078084804

Offset: 0

Author

Alois P. Heinz, Apr 13 2024

Keywords

• nonn
• more

Examples

			a(3) = 3 + 2 + 0 + 0 + 0 = 5:
   1
  _2   /\      _2           1           1
  _2  /  \      3  /\/\     3  /\       3    /\     3
  _2 /    \    _2 /    \    3 /  \/\    3 /\/  \    4 /\/\/\    .
		

Links

• Wikipedia, Counting lattice paths

Crossrefs

Column k=2 of A371928.

Cf. A000108, A051485, A152880.

Programs

Maple

g:= proc(x, y, p) (h-> `if`(x=0, add(`if`(coeff(h, z, i)=2, 1, 0),
      i=0..degree(h)), b(x, y, h)))(p+`if`(coeff(p, z, y)<3, z^y, 0))
    end:
b:= proc(x, y, p) option remember; `if`(y+2<=x,
      g(x-1, y+1, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
    end:
a:= n-> g(2*n, 0$2):
seq(a(n), n=0..18);