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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.

%I A357652 #12 Oct 12 2022 15:22:54
%S A357652 1,1,3,21,147,1323,12618,131085,1430187,16297347,191987562,2325379147,
%T A357652 28821761290,364290802138,4682375323044,61067639131197,
%U A357652 806671205158587,10776418254992139,145413196382253114,1979833455619072515,27174458892459331530,375722890152963114330
%N 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.
%H A357652 Alois P. Heinz, <a href="/A357652/b357652.txt">Table of n, a(n) for n = 0..839</a>
%H A357652 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A357652 a(n) = (A001246(n) + A129123(n))/2 = (A000108(n)^2 + A129123(n))/2.
%p A357652 b:= proc(n) option remember; `if`(n<2, 1, (2*n*(90*n^5-309*n^4+147*n^3+
%p A357652       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 A357652        b(n-2))/(n*(n+1)^3*(15*n^2-34*n+7)))
%p A357652     end:
%p A357652 a:= n-> ((binomial(n+n, n)/(n+1))^2+b(n))/2:
%p A357652 seq(a(n), n=0..21);
%Y A357652 Cf. A000108, A001246, A129123, A355481.
%K A357652 nonn
%O A357652 0,3
%A A357652 _Alois P. Heinz_, Oct 07 2022