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A225824
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The number of increasing rooted identity trees.
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1
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1, 1, 1, 4, 8, 38, 206, 1200, 7034, 53012, 465190, 4072948, 40967916, 438348328, 5113450320, 63135973560, 835727519000, 11736948927176, 175225673352928, 2749604628466960, 45540211979269216, 791473522065224592, 14405894145521294480, 274114459633006310336
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OFFSET
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1,4
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COMMENTS
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Unlabeled rooted identity trees are counted by A004111. An increasing tree is a labeled tree such that any path from the root to each node is increasing.
The first 8 terms are A032301(1..8). The 9th term is A032301(9) plus the 210 increasing labelings of the tree shown in the example.
The number of increasing labelings of a tree is n! divided by the product over all nodes of the number of descendants of the node (including the node itself). For the tree shown in the example, the number of labelings is 9!/(1 * 2 * 3 * 4 * 9 * 1 * 2 * 4 * 1) = 210. - Andrew Howroyd, Feb 02 2021
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LINKS
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FORMULA
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a(n) = n!*W(1,n) where W(d,r) = (1/r)^d * [x^r] x*exp(Sum_{i>=1} Sum_{j>=1} (-1)^(i+1)*W(i*d, j)*x^(i*j)/i). - Andrew Howroyd, Jan 22 2021
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EXAMPLE
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There are 210 increasing labelings for this identity tree:
..........0............
........./ \...........
........0 0..........
......./ / \.........
......0 0 0........
...../ | .......
....0 0 .......
.../ .......
..0 .......
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PROG
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(PARI) seq(n)={my(v=vector(n)); for(n = 1, #v, fordiv(n, d, my(r=n/d); v[d] += x^r*polcoef(exp(sum(i=1, r-1, (-1)^(i+1)*subst(v[i*d], x, x^i)/i) + O(x^r)), r-1)/r^d )); Vec(serlaplace(v[1]+O(x*x^n)))} \\ Andrew Howroyd, Jan 22 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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