A261785 - OEIS

%I #14 Jun 10 2017 10:31:11

%S 1,1,4,13,101,571,6735,54713,873019,9274471,187278048,2460190261,

%T 60205154959,942541045811,27121249048036,492972449490417,

%U 16312991079531595,337650093459084079,12633283010644517490,293339323822142071021,12245145846336974734339

%N Sum over all Motzkin paths of length n of products over all peaks p of (n*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p.

%H Alois P. Heinz, <a href="/A261785/b261785.txt">Table of n, a(n) for n = 0..400</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>

%F a(n) = A258309(n,n).

%p b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,

%p       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)

%p                   +b(x-1, y, false, k) +b(x-1, y+1, true, k)))

%p     end:

%p a:= n-> b(n, 0, false, n):

%p seq(a(n), n=0..25);

%t b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];

%t a[n_] := b[n, 0, False, n];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 10 2017, translated from Maple *)

%Y Main diagonal of A258309.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 31 2015