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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. 10
1, 1, 3, 12, 58, 321, 1975, 13265, 96073, 743753, 6113769, 53086314, 484861924, 4641853003, 46441475253, 484327870652, 5252981412262, 59132909030463, 689642443691329, 8319172260103292, 103645882500123026, 1331832693574410475, 17629142345935969713 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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.

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

MAPLE

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

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

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

    end:

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

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 *)

CROSSREFS

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

Sequence in context: A291488 A090363 A115086 * A196708 A184511 A125276

Adjacent sequences:  A258170 A258171 A258172 * A258174 A258175 A258176

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 22 2015

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)