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A258312 Sum over all Motzkin paths of length 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 A258312 #14 Jun 10 2017 10:32:01
%S A258312 1,1,2,5,14,43,141,490,1785,6789,26809,109632,462755,2012441,8997402,
%T A258312 41297927,194306557,936082502,4612095475,23219012907,119328025012,
%U A258312 625545408219,3342370197206,18190297736313,100768960522871,567886743369378,3253833477309093
%N A258312 Sum over all Motzkin paths of length n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p.
%H A258312 Alois P. Heinz, <a href="/A258312/b258312.txt">Table of n, a(n) for n = 0..800</a>
%H A258312 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%p A258312 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258312       `if`(x=0, 1, b(x-1, y-1, false) *`if`(t, x/y, 1)
%p A258312                   +b(x-1, y, false)+b(x-1, y+1, true)))
%p A258312     end:
%p A258312 a:= n-> b(n, 0, false):
%p A258312 seq(a(n), n=0..30);
%t A258312 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, False] + b[x-1, y+1, True]]];
%t A258312 a[n_] := b[n, 0, False];
%t A258312 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 10 2017, translated from Maple *)
%Y A258312 Column k=0 of A258306 and A258307.
%K A258312 nonn
%O A258312 0,3
%A A258312 _Alois P. Heinz_, May 25 2015