OFFSET
1,13
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
EXAMPLE
The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are:
: o : o o : o : o :
: /( )\ : / \ | : | : | :
: o o o o : o o o : o : o :
: : | | /|\ : | : | :
: : o o o o o : o : o :
: : : / \ : | :
: : : o o : o :
: : : : | :
: : : : o :
: : : : :
: ---1--- : -----2----- : --3-- : -4- :
Thus row 5 = [0, 1, 2, 1, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 1, 1;
0, 1, 2, 1, 1;
0, 1, 2, 2, 1, 1;
0, 1, 4, 3, 2, 1, 1;
0, 1, 4, 5, 3, 2, 1, 1;
0, 1, 7, 7, 6, 3, 2, 1, 1;
0, 1, 8, 12, 8, 6, 3, 2, 1, 1;
0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1;
0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
...
MAPLE
with(numtheory):
T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
add(add(`if`(d<k, 0, T(d, k-1)*d), d=divisors(j))*
T(n-j, k), j=1..n-1)/(n-1)))
end:
seq(seq(T(n, k), k=0..n-1), n=1..14);
MATHEMATICA
T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[d<k, 0, T[d, k-1]*d], {d, Divisors[j]}] * T[n-j, k], {j, 1, n-1}]/(n-1)]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 08 2014
STATUS
approved