A121320 Number of vertices in all ordered (plane) trees with n edges that are at distance two from all the leaves above them.
0, 0, 1, 2, 6, 18, 59, 203, 724, 2643, 9802, 36755, 138935, 528406, 2019419, 7748125, 29825844, 115132729, 445498768, 1727434607, 6710501025, 26110567532, 101744332967, 396983837719, 1550777652546, 6064476854065, 23739056348161
Offset: 0
Examples
a(4)=6 since the root has the distance two property for the trees uudduudd and uudududd. There are similar points at height 1 for uuududdd, uuudddud and uduuuddd. The distance two point is at height 2 for uuuudddd.
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..500
Programs
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Mathematica
CoefficientList[Series[x^2(1 + 1/Sqrt[1 - 4x])/(2(1 - x - x^2)), {x, 0, 26}], x] (* Robert G. Wilson v, Aug 21 2006 *) -
PARI
seq(n)={Vec(x^2*(1 + 1/sqrt(1 - 4*x + O(x^(n-1))))/(2 - 2*x - 2*x^2), -(n+1))} \\ Andrew Howroyd, Apr 06 2020
Formula
G.f.: x^2*(1 + 1/sqrt(1 - 4*x))/(2 - 2*x - 2*x^2). - Reformulated by Georg Fischer, Apr 06 2020
Conjecture: (-n+2)*a(n) +(5*n-12)*a(n-1) +(-3*n+8)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 22 2016
a(n) ~ 2^(2*n-1) / (11*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 07 2020
Extensions
More terms from Robert G. Wilson v, Aug 21 2006