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Search: id:A118376
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| A118376 |
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Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node. |
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1
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| 1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, 4378663296, 27081760768, 168530142720, 1054464293888, 6629484729344
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The number of trees with leaf nodes equal to 1 is counted by the sequence A001003 of super-Catalan numbers. The number of binary trees is counted by the sequence A007317 and the number of binary trees with leaf nodes equal to 1 is counted by the sequence A000108 of Catalan numbers.
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REFERENCES
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J. R. Johnson, M. P\"{u}schel, In search for the optimal Walsh-Hadamard transform, Proc. ICASSP, Vol. 4, 2000, pp. 3347-3350.
Pawel Hitczenko, Jeremy R. Johnson, Hung-Jen Huang, Distribution of a class of divide and conquer recurrences arising from the computation of the Walsh-Hadamard transform, Theoretical Computer Science, Vol. 352, 2006, pp. 8-30.
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FORMULA
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Recurrence: T(1) = 1; For n > 1, T(n) = 1 + sum_{n=n1+...+nt} T(n1)*...*T(nt) G.F.: (-1+(1-8*z+8*z^2)^(1/2))/(-4+4*z)
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EXAMPLE
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T(3) = 6 because there are six trees
3 3 3 3 3 3
2 1 2 1 1 2 1 2 1 1 1
1 1 1 1
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MAPLE
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T := proc(n) option remember; local C, s, p, tp, k, i; if n = 1 then return 1; else s := 1; for k from 2 to n do C := combinat[composition](n, k); for p in C do tp := map(T, p); s := s + mul(tp[i], i=1..nops(tp)); end do; end do; end if; return s; end;
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CROSSREFS
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Cf. A001003, A007317, A000108.
Sequence in context: A138020 A046646 A152322 this_sequence A085486 A152318 A152328
Adjacent sequences: A118373 A118374 A118375 this_sequence A118377 A118378 A118379
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KEYWORD
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nonn
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AUTHOR
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Jeremy Johnson (jjohnson(AT)cs.drexel.edu), May 15 2006
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