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A126324 a(2n) = Cat(n), a(2n+1) = 3*Cat(n), where Cat(n) = binomial(2n,n)/(n+1) are the Catalan numbers (A000108). 2
1, 3, 1, 3, 2, 6, 5, 15, 14, 42, 42, 126, 132, 396, 429, 1287, 1430, 4290, 4862, 14586, 16796, 50388, 58786, 176358, 208012, 624036, 742900, 2228700, 2674440, 8023320, 9694845, 29084535, 35357670, 106073010, 129644790, 388934370, 477638700 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of hex trees with n edges and n branches (i.e. each branch consists of a single edge). A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). a(n)=A126321(n,n). a(2n)=A000108(n), a(2n+1)=3*A000108(n).

LINKS

Robert Israel, Table of n, a(n) for n = 0..3334

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

FORMULA

G.f.: (1+3*z)*(1-sqrt(1-4*z^2))/(2*z^2). [corrected by Robert Israel, Dec 29 2016]

Conjecture: (5*n+17)*(n+2)*a(n) -36*a(n-1) -4*(5*n+22)*(n-2)*a(n-2)=0. - R. J. Mathar, Jun 17 2016

Conjecture confirmed, because the g.f. satisfies the d.e. (-36*z+34)*g(z)+(-148*z^3+32*z)*g'(z)+(-20*z^4+5*z^2)*g''(z)-162*z-34 = 0. - Robert Israel, Dec 29 2016

MAPLE

c:=n->binomial(2*n, n)/(n+1): a:=proc(n) if n mod 2=0 then c(n/2) else 3*c((n-1)/2) fi end: seq(a(n), n=0..41);

MATHEMATICA

CoefficientList[Series[(1+3*x)*(1-Sqrt[1-4*x^2])/(2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 23 2018 *)

PROG

(PARI) x='x+O('x^50); Vec((1+3*x)*(1-sqrt(1-4*x^2))/(2*x^2)) \\ G. C. Greubel, Oct 23 2018

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+3*x)*(1-Sqrt(1-4*x^2))/(2*x^2))); // G. C. Greubel, Oct 23 2018

CROSSREFS

Cf. A000108, A126321.

Sequence in context: A126088 A294248 A085671 * A035557 A305735 A248947

Adjacent sequences:  A126321 A126322 A126323 * A126325 A126326 A126327

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 25 2006

STATUS

approved

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Last modified July 14 09:22 EDT 2020. Contains 335720 sequences. (Running on oeis4.)