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A034854
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Triangle giving number of labeled trees with n >= 3 nodes and diameter d >= 2.
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3
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3, 4, 12, 5, 60, 60, 6, 210, 720, 360, 7, 630, 6090, 7560, 2520, 8, 1736, 47040, 112560, 80640, 20160, 9, 4536, 363384, 1496880, 1829520, 907200, 181440, 10, 11430, 2913120, 19207440, 36892800, 28274400, 10886400, 1814400
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OFFSET
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0,1
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LINKS
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FORMULA
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Reference gives recurrence.
Let G_k(x) be the e.g.f. counting the number of rooted labeled trees with height <= k. Then G_k(x) is defined recursively by G_0(x) = x, G_k(x) = x*exp(G_{k-1}(x). Let H_k(x) be the e.g.f. counting rooted labeled trees of height k. Then H_0(x) = x, H_k(x) = G_k(x) - G_{k-1}(x) for k >= 1. The e.g.f. for column d=2m+1 is H_m(x)^2/2. The e.g.f. for column d=2m is G_{m-1}(x)*(exp(H_{m-1}(x)) - 1 - H_{m-1}(x)). (End)
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EXAMPLE
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Triangle begins:
3;
4, 12;
5, 60, 60;
6, 210, 720, 360;
7, 630, 6090, 7560, 2520;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004
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STATUS
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approved
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