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URL: https://oeis.org/A228343

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A228343
The number of ordered trees with bicolored single edges on the boundary.
0
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
0,2
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.
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
EXAMPLE
When n=3 the five trees contribute as follows: UUUDDD 8; UUDDUD, UDUUDD,UUDUDD 2 each; and UDUDUD just 1.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A149948 A093129 A020876 * A149949 A149950 A370801
KEYWORD
nonn
AUTHOR
Louis Shapiro, Aug 20 2013
STATUS
approved