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

0,2

Links

Table of n, a(n) for n=0..27.

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

Cf. A000108, A228197.

Sequence in context: A149948 A093129 A020876 * A149949 A149950 A370801

Adjacent sequences: A228340 A228341 A228342 * A228344 A228345 A228346

Keyword

nonn

Author

Louis Shapiro, Aug 20 2013

Status

approved