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A292694 Sum over all Dyck paths of semilength n of products over all peaks p of (n*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p.

%I A292694 #13 Jun 02 2018 10:38:00
%S A292694 1,2,24,764,47650,4953222,776036520,171140340632,50569280587134,
%T A292694 19291547098210250,9231053150452094896,5414004448824367167444,
%U A292694 3819333773584571070766756,3190486349393577447421521614,3114480787139044226695876470000,3512892958123523912923517986350000
%N A292694 Sum over all Dyck paths of semilength n of products over all peaks p of (n*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p.
%H A292694 Alois P. Heinz, <a href="/A292694/b292694.txt">Table of n, a(n) for n = 0..200</a>
%H A292694 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%F A292694 a(n) = A258222(n,n).
%p A292694 b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p A292694       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
%p A292694                  + b(x-1, y+1, true, k)  ))
%p A292694     end:
%p A292694 a:= n-> b(2*n, 0, false, n):
%p A292694 seq(a(n), n=0..18);
%t A292694 b[x_, y_, t_, k_] := b[x, y, t, k] = If[y>x || y<0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k] If[t, (k*x + y)/y, 1] + b[x - 1, y + 1, True, k]]];
%t A292694 a[n_] := b[2n, 0, False, n];
%t A292694 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A292694 Main diagonal of A258222.
%K A292694 nonn
%O A292694 0,2
%A A292694 _Alois P. Heinz_, Sep 20 2017