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Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.
10

%I #17 Feb 06 2017 14:04:14

%S 1,1,5,89,5933,1540161,1584150165,6497470064169,106497075348688637,

%T 6980195689972655145233,1829876050804408046228327525,

%U 1918781572083632396857805205324025,8047973452254281276702044410544321359565,135022681866797995009325363468217320506328688097

%N Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

%C 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 Alois P. Heinz, <a href="/A258181/b258181.txt">Table of n, a(n) for n = 0..55</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

%F a(n) ~ c * 2^(n*(n-1)), where c = 1.47818066525747143617276638534... . - _Vaclav Kotesovec_, Jun 01 2015

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

%p `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, 2^(x-y), 1) +

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

%p end:

%p a:= n-> b(2*n, 0, false):

%p seq(a(n), n=0..15);

%t 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, 2^(x - y), 1] + b[x - 1, y + 1, True]]];

%t a[n_] := b[2*n, 0, False];

%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)

%Y Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180.

%K nonn

%O 0,3

%A _Alois P. Heinz_, May 22 2015