

A228197


Number of nedge ordered trees with bicolored boundary edges.


2



1, 2, 8, 36, 160, 692, 2928, 12200, 50304, 205940, 838928, 3405496, 13788736, 55723592, 224863712, 906365136, 3649978880, 14687731572, 59067989072, 237424661016, 953914608320, 3831159414552, 15381896102432, 61739966366256, 247750559632640, 993955865320392, 3986890331450528
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OFFSET

0,2


LINKS

Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.


FORMULA

G.f.: (1+4*x^2*B^2*C)/(12*x), C is the Catalan g.f. (see A000108) and B =(14*x)^(1/2) is the g.f. for the central binomial coefficients (A000984).
Conjecture: (n+1)*a(n) +2*(5*n8)*a(n1) +4*(8*n+17)*a(n2) +16*(2*n5)*a(n3)=0.  R. J. Mathar, Aug 25 2013
a(n) = 2^(2*n)2^n*JacobiP(n1,1/2,n,3) = 2^(2*n)2*A082590(n1), which satisfies the above conjecture.  Benedict W. J. Irwin, Sep 16 2016


EXAMPLE

When n=3 edges there are A000108(3)= 5 ordered trees. Four of these consist of three boundary edges each contributing 2^3 trees to the count. The last, UDUDUD, has two boundary edges giving the last 2^2 trees for a total of 36.


MATHEMATICA

CoefficientList[Series[(12*x2*x*Sqrt[14*x])/((4*x1)*(2*x1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 23 2013 *)


PROG

(PARI)
x = 'x + O('x^66);
C = serreverse( x/( 1/(1x) ) ) / x; \\ Catalan A000108
B = (14*x)^(1/2); \\ central binomial coefficients
gf = (1+4*x^2*B^2*C)/(12*x);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



