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A266386 Sum over all Motzkin paths of length n of products over all peaks p of (x_p+n*y_p)/y_p, where x_p and y_p are the coordinates of peak p.

%I A266386 #11 Jun 10 2017 14:08:13
%S A266386 1,1,4,11,62,243,1575,7721,54985,316407,2427309,15798261,129072167,
%T A266386 927577835,8008756470,62499194297,567017727805,4747097031375,
%U A266386 45051331382395,400942371431173,3965769826314532,37252002703698003,382848953452815450,3774255187367667473
%N A266386 Sum over all Motzkin paths of length n of products over all peaks p of (x_p+n*y_p)/y_p, where x_p and y_p are the coordinates of peak p.
%H A266386 Alois P. Heinz, <a href="/A266386/b266386.txt">Table of n, a(n) for n = 0..500</a>
%H A266386 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A266386 a(n) = A258306(n,n).
%p A266386 b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p A266386       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
%p A266386                   +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
%p A266386     end:
%p A266386 a:= n-> b(n, 0, false, n):
%p A266386 seq(a(n), n=0..30);
%t A266386 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, (x + k*y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
%t A266386 a[n_] := b[n, 0, False, n];
%t A266386 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 10 2017, translated from Maple *)
%Y A266386 Main diagonal of A258306.
%K A266386 nonn
%O A266386 0,3
%A A266386 _Alois P. Heinz_, Dec 28 2015