The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A228197 Number of n-edge 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 D. E. Davenport, L. K. Pudwell, L. W. Shapiro, L. C. Woodson, The Boundary of Ordered Trees, 2014. 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. E. Deutsch, L. W. Shapiro, A bijection between ordered trees and 2-Motzkin paths and its many consequences, Disc. Math. 256 (2002) 655-670. FORMULA G.f.: (1+4*x^2*B^2*C)/(1-2*x), C is the Catalan g.f. (see A000108) and B =(1-4*x)^(-1/2) is the g.f. for the central binomial coefficients (A000984). a(n) ~ 4^n * (1-1/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 23 2013 Conjecture: (-n+1)*a(n) +2*(5*n-8)*a(n-1) +4*(-8*n+17)*a(n-2) +16*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 25 2013 a(n) = 2^(2*n)-2^n*JacobiP(n-1,1/2,-n,3) = 2^(2*n)-2*A082590(n-1), 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[(1-2*x-2*x*Sqrt[1-4*x])/((4*x-1)*(2*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 23 2013 *) Table[2^(2n)-2^n*JacobiP[n-1, 1/2, -n, 3], {n, 0, 20}] (* Benedict W. J. Irwin, Sep 16 2016 *) PROG (PARI) x = 'x + O('x^66); C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108 B = (1-4*x)^(-1/2); \\ central binomial coefficients gf = (1+4*x^2*B^2*C)/(1-2*x); Vec(gf) \\ Joerg Arndt, Aug 21 2013 CROSSREFS Cf. A000108, A000984, A228178, A228180. Sequence in context: A321110 A228791 A088675 * A326244 A027743 A152124 Adjacent sequences: A228194 A228195 A228196 * A228198 A228199 A228200 KEYWORD nonn AUTHOR Louis Shapiro, Aug 20 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 23 16:34 EDT 2023. Contains 365554 sequences. (Running on oeis4.)