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A228343
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The number of ordered trees with bicolored single edges on the boundary.
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0
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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
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OFFSET
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0,2
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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When n=3 the five trees contribute as follows: UUUDDD 8; UUDDUD, UDUUDD,UUDUDD 2 each; and UDUDUD just 1.
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MATHEMATICA
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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 *)
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PROG
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(PARI)
x = 'x + O('x^66);
C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
gf = (1+x^2*C^5)/(1-2*x);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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