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 A102892 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the number of edges from the root to the first branch point is k. 3
 1, 0, 1, 1, 0, 2, 5, 3, 0, 4, 25, 16, 6, 0, 8, 130, 83, 32, 12, 0, 16, 700, 442, 166, 64, 24, 0, 32, 3876, 2420, 884, 332, 128, 48, 0, 64, 21945, 13566, 4840, 1768, 664, 256, 96, 0, 128, 126500, 77539, 27132, 9680, 3536, 1328, 512, 192, 0, 256 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The statistic "number of edges from the root to the first branchpoint" is equal to 0 if root is a branchpoint and it is equal to the total number of edges if there is no branchpoint. Row n contains n+1 terms. Row sums yield the ternary numbers (A001764). Column 0 yields A102893. Column 1 yields A030983. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1274 M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998. FORMULA T(n, 0) = 5*binomial(3n-1, n-2)/(3n-1) for n > 0. T(n, k) = [2^(k-1)/(n-k+1)]binomial(3n-3k+1, n-k)-[2^k/(n-k)]binomial(3n-3k-2, n-k-1) for 0 < k < n. T(n, n) = 2^(n-1) (n > 0). G.f.: (1/2)*g(2-g) + g^2*(1-2*z)/(2*(1-2*t*z)), where g = 1 + z*g^3 is the g.f. of the ternary numbers (A001764). EXAMPLE T(2,0)=1 because we have /\ and T(2,2)=2 because we have /_ and _\. Triangle starts:     1;     0,  1;     1,  0,  2;     5,  3,  0,  4;    25, 16,  6,  0, 8;   130, 83, 32, 12, 0, 16;   ... MAPLE T:=proc(n, k) if n=0 and k=0 then 1 elif k=0 then 5*binomial(3*n-1, n-2)/(3*n-1) elif k

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Last modified May 25 19:13 EDT 2019. Contains 323576 sequences. (Running on oeis4.)