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A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.

%I A258175 #16 Feb 06 2017 14:00:27
%S A258175 1,2,12,114,1448,22770,424164,9095450,220023184,5914998594,
%T A258175 174682531260,5614908340866,194967208057272,7267467723747218,
%U A258175 289270983756577620,12239218862861690250,548301077168477951520,25918121712918957399426,1288797080051656060595820
%N A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.
%C A258175 A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.
%H A258175 Alois P. Heinz, <a href="/A258175/b258175.txt">Table of n, a(n) for n = 0..350</a>
%H A258175 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258175 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258175       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x+y, 1) +
%p A258175                    b(x-1, y+1, true)  ))
%p A258175     end:
%p A258175 a:= n-> b(2*n, 0, false):
%p A258175 seq(a(n), n=0..20);
%t A258175 b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x + y, 1] + b[x - 1, y + 1, True]]];
%t A258175 a[n_] := b[2*n, 0, False];
%t A258175 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258175 Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258176, A258177, A258178, A258179, A258180, A258181.
%K A258175 nonn
%O A258175 0,2
%A A258175 _Alois P. Heinz_, May 22 2015