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A292085 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. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,13
LINKS
FORMULA
A(n,k) = Sum_{j=1..k} A292086(n,j).
EXAMPLE
: 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, ...
MAPLE
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);
MATHEMATICA
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]}]];
Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)
CROSSREFS
Main diagonal gives A000669.
Sequence in context: A283308 A339959 A255636 * A262163 A293112 A306910
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 08 2017
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)