A258173 Sum over all Dyck paths of semilength n of products over all peaks p of y_p, where y_p is the y-coordinate of peak p.

1, 1, 3, 12, 58, 321, 1975, 13265, 96073, 743753, 6113769, 53086314, 484861924, 4641853003, 46441475253, 484327870652, 5252981412262, 59132909030463, 689642443691329, 8319172260103292, 103645882500123026, 1331832693574410475, 17629142345935969713

Offset: 0

Author

Alois P. Heinz, May 22 2015

Keywords

• nonn

Comments

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.

Number of general rooted ordered trees with n edges and "back edges", which are additional edges connecting vertices to their ancestors. Every vertex specifies an ordering on the edges to its children and back edges to its ancestors altogether; it may be connected to the same ancestor by multiple back edges, distinguishable only by their relative ordering under that vertex. - Li-yao Xia, Mar 06 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
• Antti Karttunen, Bijection between rooted trees with back edges and Dyck paths with multiplicity, SeqFans mailing list, Mar 2 2017.
• Wikipedia, Lattice path
• Li-yao Xia, Definition and enumeration of rooted trees with back edges in Haskell, blog post, Mar 1 2017.

Crossrefs

Cf. A000108, A000698, A005411, A005412, A258172, A258174, A258175, A258176, A258177, A258178, A258179, A258180, A258181, A371567.

Programs

Maple

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
     `if`(x=0, 1, b(x-1, y-1, 0)*y^t+b(x-1, y+1, 1)))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);

Mathematica

nmax = 25; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x/(k*x + 2*x - 1/g[k+1]); CoefficientList[Series[g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2015, after Sergei N. Gladkovskii *)

Formula

G.f.: T(0), where T(k) = 1 - x/(k*x + 2*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2015

Conjecture: a(n) = A371567(n-1,0) for n > 0 with a(0) = 1. - Mikhail Kurkov, Nov 07 2024