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A292085
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Number A(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes or outdegrees larger than k; square array A(n,k), n>=0, k>=0, 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, 4, 3, 0, 1, 1, 2, 5, 9, 6, 0, 1, 1, 2, 5, 11, 23, 11, 0, 1, 1, 2, 5, 12, 30, 58, 23, 0, 1, 1, 2, 5, 12, 32, 80, 156, 46, 0, 1, 1, 2, 5, 12, 33, 87, 228, 426, 98, 0, 1, 1, 2, 5, 12, 33, 89, 251, 656, 1194, 207, 0
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OFFSET
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1,13
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LINKS
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FORMULA
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A(n,k) = Sum_{j=1..k} A292086(n,j).
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EXAMPLE
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: T(4,3) = 4 :
: :
: o o o o :
: / \ / \ / \ /|\ :
: o N o o o N o N N :
: / \ ( ) ( ) /|\ ( ) :
: o N N N N N N N N N N :
: ( ) :
: N N :
: :
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, ...
0, 2, 4, 5, 5, 5, 5, 5, ...
0, 3, 9, 11, 12, 12, 12, 12, ...
0, 6, 23, 30, 32, 33, 33, 33, ...
0, 11, 58, 80, 87, 89, 90, 90, ...
0, 23, 156, 228, 251, 258, 260, 261, ...
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MAPLE
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b:= proc(n, i, v, k) option remember; `if`(n=0,
`if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
`if`(v=n, 1, add(binomial(A(i, k)+j-1, j)*
b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
end:
A:= proc(n, k) option remember; `if`(n<2, n,
add(b(n, n+1-j, j, k), j=2..min(n, k)))
end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);
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MATHEMATICA
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b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
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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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