cp's OEIS Frontend

A258176 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 A258176 #15 Feb 06 2017 14:01:09
%S A258176 1,1,7,142,9354,2503597,3260627607,24105227716863,1028836978599566213,
%T A258176 290383808553140390346475,511963364817949502725911280781,
%U A258176 6704846980724405836568589845161191576,584709361918378923208855262622537662297053728
%N A258176 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 A258176 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 A258176 Alois P. Heinz, <a href="/A258176/b258176.txt">Table of n, a(n) for n = 0..45</a>
%H A258176 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258176 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258176       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^y, 1) +
%p A258176                    b(x-1, y+1, true)  ))
%p A258176     end:
%p A258176 a:= n-> b(2*n, 0, false):
%p A258176 seq(a(n), n=0..15);
%t A258176 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 A258176 a[n_] :=  b[2*n, 0, False];
%t A258176 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258176 Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258177, A258178, A258179, A258180, A258181.
%K A258176 nonn
%O A258176 0,3
%A A258176 _Alois P. Heinz_, May 22 2015