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A258174 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 A258174 #15 Feb 06 2017 13:59:59
%S A258174 1,1,7,84,1486,35753,1111931,43150593,2035666985,114412223081,
%T A258174 7538224510181,574552299138202,50096579094908148,4949493445607316419,
%U A258174 549534510282406667069,68071071679372210762156,9347203754680124767253730,1414740620049957735248175695
%N A258174 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 A258174 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 A258174 Alois P. Heinz, <a href="/A258174/b258174.txt">Table of n, a(n) for n = 0..250</a>
%H A258174 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%p A258174 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258174       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x*y, 1) +
%p A258174                    b(x-1, y+1, true)  ))
%p A258174     end:
%p A258174 a:= n-> b(2*n, 0, false):
%p A258174 seq(a(n), n=0..20);
%t A258174 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 A258174 a[n_] := b[2*n, 0, False];
%t A258174 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y A258174 Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258175, A258176, A258177, A258178, A258179, A258180, A258181.
%K A258174 nonn
%O A258174 0,3
%A A258174 _Alois P. Heinz_, May 22 2015