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A258172 Sum over all Dyck paths of semilength n of products over all peaks p of x_p, where x_p is the x-coordinate of peak p.

%I A258172 #17 Feb 06 2017 13:58:38
%S A258172 1,1,5,40,434,5901,95997,1812525,38875265,932135347,24678938063,
%T A258172 714385754446,22428656766320,758632387171075,27489135956517315,
%U A258172 1061913384743418360,43550536908458238570,1889211624465639489675,86406059558668152123975,4154647501527354507485040
%N A258172 Sum over all Dyck paths of semilength n of products over all peaks p of x_p, where x_p is the x-coordinate of peak p.
%C A258172 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 A258172 Alois P. Heinz, <a href="/A258172/b258172.txt">Table of n, a(n) for n = 0..300</a>
%H A258172 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258172 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258172       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x, 1) +
%p A258172                    b(x-1, y+1, true)  ))
%p A258172     end:
%p A258172 a:= n-> b(2*n, 0, false):
%p A258172 seq(a(n), n=0..20);
%t A258172 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, 1] + b[x - 1, y + 1, True]]];
%t A258172 a[n_] := b[2*n, 0, False];
%t A258172 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258172 Cf. A000108, A000698, A005411, A005412, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180, A258181.
%K A258172 nonn
%O A258172 0,3
%A A258172 _Alois P. Heinz_, May 22 2015