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A121320 Number of vertices in all ordered (plane) trees with n edges that are at distance two from all the leaves above them. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..500

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A000045, A024718.

Sequence in context: A150042 A036675 A227373 * A148460 A148461 A002527

Adjacent sequences:  A121317 A121318 A121319 * A121321 A121322 A121323

KEYWORD

easy,nonn

AUTHOR

Louis Shapiro, Aug 25 2006

EXTENSIONS

More terms from Robert G. Wilson v, Aug 21 2006

STATUS

approved

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Last modified May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)