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

0,4

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: A351279 A036675 A227373 * A148460 A148461 A358251

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