cp's OEIS Frontend

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

%I A355481 #48 Nov 16 2022 08:53:12
%S A355481 0,0,1,4,49,441,4806,52956,614713,7341697,90118054,1130414649,
%T A355481 14447230854,187609607862,2470253990556,32922380442828,
%U A355481 443493622670313,6031353319151961,82725531355436886,1143385727109903585,15913217995801644870,222875331740976566070
%N A355481 Number of pairs of Dyck paths of semilength n such that the midpoint of the first is above the midpoint of the second.
%H A355481 Alois P. Heinz, <a href="/A355481/b355481.txt">Table of n, a(n) for n = 0..839</a>
%H A355481 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A355481 a(n) = (A001246(n) - A129123(n))/2 = (A000108(n)^2 - A129123(n))/2.
%p A355481 b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
%p A355481       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)*
%p A355481        b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
%p A355481     end:
%p A355481 a:= n-> ((binomial(n+n, n)/(n+1))^2-b(n))/2:
%p A355481 seq(a(n), n=0..21);
%t A355481 A129123[n_] := Sum[(Binomial[n, k]-Binomial[n, k-1])^4, {k, 0, Floor[n/2]}];
%t A355481 a[n_] := (CatalanNumber[n]^2 - A129123[n])/2;
%t A355481 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Nov 16 2022 *)
%Y A355481 Cf. A000108, A001246, A129123, A357652.
%K A355481 nonn
%O A355481 0,4
%A A355481 _Alois P. Heinz_, Oct 07 2022