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).
LINKS
Robert Israel, Table of n, a(n) for n = 0..3334
Frank Harary and Ronald C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
FORMULA
a(n) = A126321(n,n).
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
Sum_{n>=0} a(n)/4^n = 14 - 7*sqrt(3). - Amiram Eldar, Jul 10 2023
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 *)
Table[If[EvenQ[n], CatalanNumber[n/2], 3*CatalanNumber[(n-1)/2]], {n, 0, 50}] (* Harvey P. Dale, Apr 19 2023 *)
PROG
(PARI) my(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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 25 2006
STATUS
approved