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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);

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-Fran├žois Alcover, Nov 07 2017, after Alois P. Heinz *)

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 A293112 A112185

Adjacent sequences:  A292082 A292083 A292084 * A292086 A292087 A292088

KEYWORD

nonn,tabl,changed

AUTHOR

Alois P. Heinz, Sep 08 2017

STATUS

approved

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Last modified November 19 08:42 EST 2017. Contains 294923 sequences.