A228343 The number of ordered trees with bicolored single edges on the boundary.
1, 2, 5, 15, 50, 175, 625, 2251, 8142, 29544, 107538, 392726, 1439204, 5292833, 19533241, 72333107, 268728214, 1001448308, 3742866166, 14026901282, 52701685564, 198481560878, 749170991770, 2833635556670, 10738689128460, 40770816357920, 155056284790340, 590644481896972
Offset: 0
Keywords
Examples
When n=3 the five trees contribute as follows: UUUDDD 8; UUDDUD, UDUUDD,UUDUDD 2 each; and UDUDUD just 1.
Links
- Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8; preprint, 2014.
Programs
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Mathematica
Table[FullSimplify[I*2^n - 5/2*Gamma[3+2*n] * HypergeometricPFQRegularized[{1,3/2+n,2+n},{n,5+n},2]],{n,0,20}] (* Vaclav Kotesovec, Jan 31 2014 *) -
PARI
x = 'x + O('x^66); C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108 gf = (1+x^2*C^5)/(1-2*x); Vec(gf) \\ Joerg Arndt, Aug 21 2013
Formula
G.f.: (1+x^2*C^5)/(1-2*x) where C is the Catalan number generating function (cf. A000108).
D-finite with recurrence: -(n+3)*(n-2)*a(n) +6*(n^2-2)*a(n-1) -4*n*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 25 2013
a(n) -2*a(n-1) = A000344(n). - R. J. Mathar, Aug 25 2013
a(n) ~ 5 * 2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 31 2014