A355481 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is above the midpoint of the second.

0, 0, 1, 4, 49, 441, 4806, 52956, 614713, 7341697, 90118054, 1130414649, 14447230854, 187609607862, 2470253990556, 32922380442828, 443493622670313, 6031353319151961, 82725531355436886, 1143385727109903585, 15913217995801644870, 222875331740976566070

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, A357652.

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);

Mathematica

A129123[n_] := Sum[(Binomial[n, k]-Binomial[n, k-1])^4, {k, 0, Floor[n/2]}];
a[n_] := (CatalanNumber[n]^2 - A129123[n])/2;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022 *)

Formula

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