A318799 -id:A318799

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A318753

Number A(n,k) of rooted trees with n nodes such that no more than k subtrees extending from the same node have the same number of nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

+10
13

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 3, 3, 0, 0, 1, 1, 2, 4, 7, 6, 0, 0, 1, 1, 2, 4, 8, 15, 12, 0, 0, 1, 1, 2, 4, 9, 18, 34, 25, 0, 0, 1, 1, 2, 4, 9, 19, 43, 79, 51, 0, 0, 1, 1, 2, 4, 9, 20, 46, 102, 190, 111, 0, 0, 1, 1, 2, 4, 9, 20, 47, 110, 250, 457, 240, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,19

LINKS

Alois P. Heinz, Antidiagonals n = 0..200, flattened

FORMULA

A(n,k) = Sum_{j=0..k} A318754(n,j) for n > 0.

A(n,n+j) = A000081(n) for j >= -1.

EXAMPLE

Square array A(n,k) begins:

0, 0, 0, 0, 0, 0, 0, 0, 0, ...

1, 1, 1, 1, 1, 1, 1, 1, 1, ...

0, 1, 1, 1, 1, 1, 1, 1, 1, ...

0, 1, 2, 2, 2, 2, 2, 2, 2, ...

0, 2, 3, 4, 4, 4, 4, 4, 4, ...

0, 3, 7, 8, 9, 9, 9, 9, 9, ...

0, 6, 15, 18, 19, 20, 20, 20, 20, ...

0, 12, 34, 43, 46, 47, 48, 48, 48, ...

0, 25, 79, 102, 110, 113, 114, 115, 115, ...

MAPLE

g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

binomial(A(i, k)+j-1, j)*g(n-i*j, i-1, k), j=0..min(k, n/i))))

end:

A:= (n, k)-> g(n-1$2, k):

seq(seq(A(n, d-n), n=0..d), d=0..14);

MATHEMATICA

g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*g[n - i*j, i - 1, k], {j, 0, Min[k, n/i]}]]];

A[n_, k_] := g[n - 1, n - 1, k];

Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A063524, A032305, A213920, A318797, A318798, A318799, A318800, A318801, A318802, A318803, A318804.

Rows n=0-2 give: A000004, A000012, A057427.

Main diagonal gives A000081.

Cf. A318754.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 02 2018

STATUS

approved

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