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A168506
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Number of rooted plane trees of total weight n whose nodes are themselves planted plane trees whose roots are distinguished as either red or blue, the weight of each such node being equal to the size of the corresponding planted tree.
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1
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2, 2, 8, 18, 64, 188, 656, 2154, 7584, 26276, 93904, 335156, 1214944, 4417848, 16208672, 59704858, 221215552, 822721108, 3072787920, 11514183900, 43287544160, 163193085960, 616876440160, 2337336770948, 8875826681024
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OFFSET
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2,1
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REFERENCES
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T. DeVries, A case study in bivariate singularity analysis, Contemp. Math. 520 (2010) p 61.
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LINKS
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P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 412, Example VI.10 and page 503, Example VII.20.
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FORMULA
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a(n) = Sum_{k=1..floor(n/2)} 2^k/(n-k)*binomial(2*k-2,k-1)*binomial(2*n-3*k-1,n-k-1).
a(n) ~ 4^n/(8*GAMMA(3/4)*n^(5/4)).
G.f.: (1/2)*(1-sqrt(1-4*x+4*x*sqrt(1-4*x))).
G.f.: C(2*x*C(x)), where C(x)/x is the generating function for A000108.
Conjecture: n*(n-1)*(n-6)*a(n) -2*(6*n-35)*(n-1)*(n-2)*a(n-1) +4*(8*n^3-88*n^2+305*n-330)*a(n-2) +8*(8*n^3-92*n^2+314*n-305)*a(n-3) -16*(n-5)*(4*n-19)*(4*n-17)*a(n-4) =0. - R. J. Mathar, Jul 24 2012
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EXAMPLE
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There are a(4) = 8 trees of total weight 4, which can be described as follows. There are 2 rooted plane trees of size 3 and 1 rooted plane tree of size 1. Adding/planting an extra root and coloring it red or blue yields 2*2 = 4 planted plane trees of size 4 and 2*1 = 2 planted plane trees of size 2. Then the 8 trees of weight 4 are obtained as: the 1*4 = 4 trees obtained by taking the (only) rooted plane tree on one node and replacing its node with one of the planted plane trees of size 4, and the 2*2 = 4 trees obtained by taking the (only) rooted plane tree on two nodes and replacing each node with a planted plane tree of size 2.
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MAPLE
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a:=n->sum(2^k/(n-k)*binomial(2*k-2, k-1)*binomial(2*n-3*k-1, n-k-1), k=1..floor(n/2)): seq(a(n), n=2..26);
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MATHEMATICA
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gf[x_] = (1 - Sqrt[1 - 4*x + 4*x*Sqrt[1 - 4*x]])/2;
CoefficientList[Series[gf[x], {x, 0, 26}], x][[3;; ]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Timothy DeVries (tdevries(AT)math.upenn.edu), Nov 27 2009
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STATUS
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approved
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