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

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Wikipedia, Counting lattice paths

Example

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

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);

Crossrefs

Column k=2 of A371928.

Cf. A000108, A051485, A152880.

Sequence in context: A103425 A119472 A018568 * A038375 A103043 A018601

Adjacent sequences: A371899 A371900 A371901 * A371904 A371905 A371906

Keyword

nonn,more

Author

Alois P. Heinz, Apr 13 2024

Status

approved