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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..839

Wikipedia, Counting lattice paths

Formula

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

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);

Crossrefs

Cf. A000108, A001246, A129123, A355481.

Sequence in context: A169634 A088088 A206638 * A037761 A037649 A037768

Adjacent sequences: A357649 A357650 A357651 * A357653 A357654 A357655

Keyword

nonn

Author

Alois P. Heinz, Oct 07 2022

Status

approved