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A255636
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Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.
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12
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1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 3, 4, 0, 1, 1, 2, 4, 7, 8, 0, 1, 1, 2, 4, 8, 15, 17, 0, 1, 1, 2, 4, 9, 18, 35, 36, 0, 1, 1, 2, 4, 9, 19, 43, 81, 79, 0, 1, 1, 2, 4, 9, 20, 46, 102, 195, 175, 0, 1, 1, 2, 4, 9, 20, 47, 110, 251, 473, 395, 0
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OFFSET
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1,13
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COMMENTS
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Any rootward k-node path starting at a leaf contains the root or a branching node.
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LINKS
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EXAMPLE
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: o o o o o o o o
: /(|)\ | / \ /|\ | | / \ |
: o ooo o o o o o o o o o o o o
: /( )\ /|\ / \ | / \ | |
: o o o o o o o o o o o o o o
: /|\ / \ / \ |
: o o o o o o o o
: A(6,2) = 8 / \
: o o
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, 2, ...
0, 2, 3, 4, 4, 4, 4, 4, 4, 4, ...
0, 4, 7, 8, 9, 9, 9, 9, 9, 9, ...
0, 8, 15, 18, 19, 20, 20, 20, 20, 20, ...
0, 17, 35, 43, 46, 47, 48, 48, 48, 48, ...
0, 36, 81, 102, 110, 113, 114, 115, 115, 115, ...
0, 79, 195, 251, 273, 281, 284, 285, 286, 286, ...
0, 175, 473, 625, 684, 706, 714, 717, 718, 719, ...
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MAPLE
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with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
`if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
end:
A:= (n, k)-> g(n-1, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);
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MATHEMATICA
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g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[d*(g[d - 1, k] - If[d == k, 1, 0]), {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n]; A[n_, k_] := g[n - 1, k]; Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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