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A006632 a(n) = 3*binomial(4*n-1,n-1)/(4*n-1).
(Formerly M2997)
21
1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 1111731933, 9338434700, 79155435870, 676196049060, 5815796869995, 50318860986108, 437662920058980, 3824609516638444, 33563127932394060, 295655735395397520, 2613391671568320765 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the number of ordered trees (A000108) with 3n-1 edges in which every non-leaf vertex has exactly two leaf children (no restriction on non-leaf children). For example, a(2) counts the 3 trees

\/......\/......\/

.\|/...\|/....\|/  . [David Callan, Aug 22 2014]

REFERENCES

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..22.

O. Aichholzer, A. Asinowski, T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 438

FORMULA

a(n) = binomial(4*n-1, n)/(4*n-1) = 3*binomial(4*n-2, n-1) - binomial(4*n-2, n). - David Callan, Sep 15 2004

G.f.: g^3 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

a(n) = (3/4)*binomial(4*n,n)/(4*n-1). - Bruno Berselli, Jan 17 2014

MAPLE

A006632:=n->3*binomial(4*n-1, n-1)/(4*n-1): seq(A006632(n), n=1..30); # Wesley Ivan Hurt, Oct 23 2017

MATHEMATICA

InverseSeries[Series[y*(1-y)^3, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *)

a[ n_] := If[ n < 1, 0, Binomial[4 n - 2, n - 1] / n]; (* Michael Somos, Aug 22 2014 *)

PROG

(PARI) a(n) = 3*binomial(4*n-1, n-1)/(4*n-1) \\ Felix Fröhlich, Oct 23 2017

CROSSREFS

A112385 divided by 2.

Cf. A000108, A002293.

Sequence in context: A047019 A099251 A171790 * A159928 A020018 A124553

Adjacent sequences:  A006629 A006630 A006631 * A006633 A006634 A006635

KEYWORD

nonn,easy

AUTHOR

Simon Plouffe

STATUS

approved

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Last modified December 17 02:54 EST 2018. Contains 318192 sequences. (Running on oeis4.)