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 A168506 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. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 REFERENCES T. DeVries, A case study in bivariate singularity analysis, Contemp. Math. 520 (2010) p 61. LINKS P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 412, Example VI.10 and page 503, Example VII.20. FORMULA 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 EXAMPLE 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. MAPLE 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); MATHEMATICA gf[x_] = (1 - Sqrt[1 - 4*x + 4*x*Sqrt[1 - 4*x]])/2; CoefficientList[Series[gf[x], {x, 0, 26}], x][[3;; ]] (* Jean-François Alcover, Jun 20 2011, after g.f. *) CROSSREFS Cf. A000108 Sequence in context: A194588 A175395 A169888 * A208966 A067640 A098277 Adjacent sequences:  A168503 A168504 A168505 * A168507 A168508 A168509 KEYWORD nonn AUTHOR Timothy DeVries (tdevries(AT)math.upenn.edu), Nov 27 2009 STATUS approved

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Last modified July 6 06:05 EDT 2020. Contains 335476 sequences. (Running on oeis4.)