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 A117357 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). 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS 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] LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 FORMULA If a(n) is the equivalent of this sequence with the root node considered to be at level k, then a(n) is the Euler transform of a(n) shifted right k places. To compute N terms, take k so that (k+1)*(k+2)/2 > N, approximate a(n) by 1 if n=k, 0 otherwise and apply this rule repeatedly. Formula from Christian G. Bower (bowerc(at)usa.net). EXAMPLE 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). MAPLE 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 CROSSREFS Cf. A117356, A000081. Sequence in context: A064650 A174619 A130083 * A029020 A035380 A036823 Adjacent sequences:  A117354 A117355 A117356 * A117358 A117359 A117360 KEYWORD nonn,changed AUTHOR Franklin T. Adams-Watters, Mar 09 2006 STATUS approved

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Last modified May 22 16:52 EDT 2013. Contains 225553 sequences.