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A117357
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Number of rooted trees with total weight n, where the weight of a node at height k is k (with the root considered to be at level 1).
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5
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0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 7, 11, 12, 16, 19, 25, 29, 38, 46, 59, 72, 91, 110, 141, 171, 214, 264, 331, 405, 509, 623, 777, 957, 1189, 1462, 1822, 2235, 2774, 3418, 4228, 5205, 6442, 7922, 9793, 12053, 14870, 18298, 22572, 27747, 34203
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OFFSET
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0,10
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COMMENTS
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Equivalently, number of trees total weight n when the weight of each node is the size of its subtree. To get the equivalence, simply distribute the weights on each node one each to the node and each of its ancestors. [From Franklin T. Adams-Watters, Oct 03 2009]
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..500
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FORMULA
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If a<k>(n) is the equivalent of this sequence with the root node considered to be at level k, then a<k>(n) is the Euler transform of a<k+1>(n) shifted right k places. To compute N terms, take k so that (k+1)*(k+2)/2 > N, approximate a<k>(n) by 1 if n=k, 0 otherwise and apply this rule repeatedly. Formula from Christian G. Bower (bowerc(at)usa.net).
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EXAMPLE
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a(9) = 2; there is one tree with root at height 1 and 4 nodes at height 2 (1+4*2 = 9) and one with root at height 1, 1 node at height 2 and 2 nodes at height 3 (1+2+2*3 = 9).
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MAPLE
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g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(g(i-k, i-k, k+1)+j-1, j)*g(n-i*j, i-1, k), j=0..n/i)))
end:
a:= n-> g(n-1, n-1, 2):
seq(a(n), n=0..60); # Alois P. Heinz, May 16 2013
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CROSSREFS
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Cf. A117356, A000081.
Sequence in context: A064650 A174619 A130083 * A029020 A035380 A036823
Adjacent sequences: A117354 A117355 A117356 * A117358 A117359 A117360
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KEYWORD
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nonn,changed
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AUTHOR
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Franklin T. Adams-Watters, Mar 09 2006
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STATUS
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approved
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