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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.

%I A258173 #45 Dec 23 2024 14:53:44
%S A258173 1,1,3,12,58,321,1975,13265,96073,743753,6113769,53086314,484861924,
%T A258173 4641853003,46441475253,484327870652,5252981412262,59132909030463,
%U A258173 689642443691329,8319172260103292,103645882500123026,1331832693574410475,17629142345935969713
%N 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.
%C A258173 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.
%C A258173 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
%H A258173 Alois P. Heinz, <a href="/A258173/b258173.txt">Table of n, a(n) for n = 0..500</a>
%H A258173 Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024. See p. 58.
%H A258173 Antti Karttunen, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-March/017314.html">Bijection between rooted trees with back edges and Dyck paths with multiplicity</a>, SeqFans mailing list, Mar 2 2017.
%H A258173 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A258173 Li-yao Xia, <a href="http://blog.poisson.chat/posts/2017-03-01-enumerating-dfs.html">Definition and enumeration of rooted trees with back edges in Haskell</a>, blog post, Mar 1 2017.
%F A258173 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
%F A258173 Conjecture: a(n) = A371567(n-1,0) for n > 0 with a(0) = 1. - _Mikhail Kurkov_, Nov 07 2024
%p A258173 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A258173      `if`(x=0, 1, b(x-1, y-1, 0)*y^t+b(x-1, y+1, 1)))
%p A258173     end:
%p A258173 a:= n-> b(2*n, 0$2):
%p A258173 seq(a(n), n=0..25);
%t A258173 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_ *)
%Y A258173 Cf. A000108, A000698, A005411, A005412, A258172, A258174, A258175, A258176, A258177, A258178, A258179, A258180, A258181, A371567.
%K A258173 nonn
%O A258173 0,3
%A A258173 _Alois P. Heinz_, May 22 2015