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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

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Index entries for sequences related to rooted trees

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);

CROSSREFS

Columns k=2-10 give: A001190, A268172, A292210, A292211, A292212, A292213, A292214, A292215, A292216.

Main diagonal gives A000669.

Cf. A244372, A288942, A292086.

Sequence in context: A286653 A283308 A255636 * A262163 A112185 A192062

Adjacent sequences:  A292080 A292081 A292082 * A292086 A292087 A292088

KEYWORD

nonn,tabl,new

AUTHOR

Alois P. Heinz, Sep 08 2017

STATUS

approved

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Last modified September 24 04:14 EDT 2017. Contains 292402 sequences.