%I #37 Apr 15 2024 18:27:04

%S 0,1,3,5,15,44,134,427,1408,4753,16321,56812,200046,711425,2551886,

%T 9222147,33544682,122712465,451169747,1666248405,6178586630,

%U 22994275870,85859249486,321562877934,1207665205311

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

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

%e a(3) = 3 + 2 + 0 + 0 + 0 = 5:

%e 1

%e _2 /\ _2 1 1

%e _2 / \ 3 /\/\ 3 /\ 3 /\ 3

%e _2 / \ _2 / \ 3 / \/\ 3 /\/ \ 4 /\/\/\ .

%p g:= proc(x, y, p) (h-> `if`(x=0, add(`if`(coeff(h, z, i)=2, 1, 0),

%p i=0..degree(h)), b(x, y, h)))(p+`if`(coeff(p, z, y)<3, z^y, 0))

%p end:

%p b:= proc(x, y, p) option remember; `if`(y+2<=x,

%p g(x-1, y+1, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)

%p end:

%p a:= n-> g(2*n, 0$2):

%p seq(a(n), n=0..18);

%Y Column k=2 of A371928.

%Y Cf. A000108, A051485, A152880.

%K nonn,more

%O 0,3

%A _Alois P. Heinz_, Apr 13 2024