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A258177 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^x_p, where x_p and y_p are the coordinates of peak p.

%I A258177 #19 Feb 06 2017 14:01:39
%S A258177 1,1,5,112,15312,22928885,475971133797,164769697242392241,
%T A258177 1674694178196441599627207,434453335415659344048321288040053,
%U A258177 2772047111897899211702422870954450438220795,919691726760748842849028933552012720445531166591469510
%N A258177 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^x_p, where x_p and y_p are the coordinates of peak p.
%C A258177 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 A258177 Alois P. Heinz, <a href="/A258177/b258177.txt">Table of n, a(n) for n = 0..35</a>
%H A258177 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258177 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258177       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, y^x, 1) +
%p A258177                    b(x-1, y+1, true)  ))
%p A258177     end:
%p A258177 a:= n-> b(2*n, 0, false):
%p A258177 seq(a(n), n=0..15);
%t A258177 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, y^x, 1] + b[x - 1, y + 1, True]]];
%t A258177 a[n_] :=  b[2*n, 0, False];
%t A258177 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258177 Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258178, A258179, A258180, A258181.
%K A258177 nonn
%O A258177 0,3
%A A258177 _Alois P. Heinz_, May 22 2015