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

0,4

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);

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 *)

Crossrefs

Cf. A000108, A001246, A129123, A357652.

Sequence in context: A045787 A067474 A053769 * A362479 A173038 A198971

Adjacent sequences: A355478 A355479 A355480 * A355482 A355483 A355484

Keyword

nonn

Author

Alois P. Heinz, Oct 07 2022

Status

approved