A357652 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is not below the midpoint of the second.

1, 1, 3, 21, 147, 1323, 12618, 131085, 1430187, 16297347, 191987562, 2325379147, 28821761290, 364290802138, 4682375323044, 61067639131197, 806671205158587, 10776418254992139, 145413196382253114, 1979833455619072515, 27174458892459331530, 375722890152963114330

Offset: 0

Author

Alois P. Heinz, Oct 07 2022

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..839
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A000108, A001246, A129123, A355481.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
      124*n^2-135*n+35)*b(n-1)+4*(n-1)^2*(4*n-5)*(4*n-3)*(15*n^2-4*n-12)*
       b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
    end:
a:= n-> ((binomial(n+n, n)/(n+1))^2+b(n))/2:
seq(a(n), n=0..21);

Formula

a(n) = (A001246(n) + A129123(n))/2 = (A000108(n)^2 + A129123(n))/2.