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A258178 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.

%I A258178 #15 Feb 06 2017 14:02:21
%S A258178 1,1,13,414,24324,2279209,311524201,58467947511,14424374692879,
%T A258178 4525566110365523,1759527523008436279,830255082140922306224,
%U A258178 467382831980334193769718,309419146352957449765072455,237980526477430552734199922151,210427994109788912088395561755374
%N A258178 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.
%C A258178 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 A258178 Alois P. Heinz, <a href="/A258178/b258178.txt">Table of n, a(n) for n = 0..200</a>
%H A258178 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258178 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258178       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^2, 1) +
%p A258178                    b(x-1, y+1, true)  ))
%p A258178     end:
%p A258178 a:= n-> b(2*n, 0, false):
%p A258178 seq(a(n), n=0..20);
%t A258178 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^2, 1] + b[x - 1, y + 1, True] ]];
%t A258178 a[n_] :=  b[2*n, 0, False];
%t A258178 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258178 Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258179, A258180, A258181.
%K A258178 nonn
%O A258178 0,3
%A A258178 _Alois P. Heinz_, May 22 2015