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A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

%I A258180 #15 Feb 06 2017 14:03:44
%S A258180 1,1,4,33,517,15326,852912,91023697,19716262702,8794395041567,
%T A258180 8016790849841585,15556074485786226848,64891787190080888991273,
%U A258180 561815453349204340865790817,10402242033224422585780623039909,423787530114579490372987256671625678
%N A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.
%C A258180 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 A258180 Alois P. Heinz, <a href="/A258180/b258180.txt">Table of n, a(n) for n = 0..75</a>
%H A258180 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258180 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258180       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, binomial(x, y), 1) +
%p A258180                    b(x-1, y+1, true)  ))
%p A258180     end:
%p A258180 a:= n-> b(2*n, 0, false):
%p A258180 seq(a(n), n=0..20);
%t A258180 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, Binomial[x, y], 1] + b[x - 1, y + 1, True]]];
%t A258180 a[n_] := b[2*n, 0, False];
%t A258180 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258180 Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258181.
%K A258180 nonn
%O A258180 0,3
%A A258180 _Alois P. Heinz_, May 22 2015