

A178520


Number of ordered trees with n edges and with no vertex of outdegree 2 that have two leaves as their two children.


2



1, 1, 1, 4, 11, 32, 98, 309, 998, 3285, 10981, 37178, 127227, 439369, 1529280, 5359314, 18894435, 66967086, 238473876, 852825314, 3061529014, 11028596473, 39853923390, 144435373636, 524837483375, 1911763716717, 6979451843306
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OFFSET

0,4


COMMENTS

Empirical: for n >= 3, a(n) is the number of Dyck npaths avoiding UUUDDD. E.g., of the 5 Dyck 3paths UUUDDD is avoided so a(3)=4.  David Scambler, Mar 24 2011


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000


FORMULA

a(n) = A178519(n,0).
G.f.: G=G(z) satisfies (G+z^2)*(1z*G)=1.
a(n) ~ sqrt((1+r)*(1r+r^2)*(15*r^3)) / (4*sqrt(Pi)*n^(3/2)*r^(n+1)), where r = 0.2587353940253970315668... is the root of the equation (1+r^3)^2 = 4*r.  Vaclav Kotesovec, Mar 21 2014


EXAMPLE

a(3)=4 because among the 5 ordered trees with 3 edges only < has a forbidden vertex (the root).


MAPLE

eq := z*G^2(1z^3)*G+1z^2; G := RootOf(eq, G): Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 0 .. 27);


MATHEMATICA

CoefficientList[Series[(1x^3Sqrt[14*x+2*x^3+x^6])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)


PROG

(PARI) x='x+O('x^50); Vec((1x^3sqrt(14*x+2*x^3+x^6))/(2*x)) \\ G. C. Greubel, Mar 24 2017


CROSSREFS

Cf. A178519.
Sequence in context: A289246 A199109 A025268 * A149232 A149233 A273038
Adjacent sequences: A178517 A178518 A178519 * A178521 A178522 A178523


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 31 2010


STATUS

approved



