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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.
13
1, 1, 3, 12, 58, 321, 1975, 13265, 96073, 743753, 6113769, 53086314, 484861924, 4641853003, 46441475253, 484327870652, 5252981412262, 59132909030463, 689642443691329, 8319172260103292, 103645882500123026, 1331832693574410475, 17629142345935969713
OFFSET
0,3
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
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
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 *)
KEYWORD
nonn,changed
AUTHOR
Alois P. Heinz, May 22 2015
STATUS
approved