OFFSET
1,13
COMMENTS
In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
EXAMPLE
The A124343(5) = 6 5-node rooted trees with thinning limbs sorted by root outdegree 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 o : o : :
: | : | : : :
: o : o : : :
: | : : : :
: o : : : :
: : : : :
: -1- : ---------2--------- : --3-- : ---4--- :
Thus row 5 = [0, 1, 3, 1, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 1, 1;
0, 1, 3, 1, 1;
0, 1, 4, 3, 1, 1;
0, 1, 9, 6, 3, 1, 1;
0, 1, 13, 13, 6, 3, 1, 1;
0, 1, 26, 25, 15, 6, 3, 1, 1;
0, 1, 42, 55, 29, 15, 6, 3, 1, 1;
0, 1, 81, 107, 68, 31, 15, 6, 3, 1, 1;
MAPLE
b:= proc(n, i, h, v) option remember; `if`(n=0,
`if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
`if`(n=v, 1, add(binomial(A(i, min(i-1, h))+j-1, j)
*b(n-i*j, i-1, h, v-j), j=0..min(n/i, v)))))
end:
A:= proc(n, k) option remember;
`if`(n<2, n, add(b(n-1$2, j$2), j=1..min(k, n-1)))
end:
T:= (n, k)-> b(n-1$2, k$2):
seq(seq(T(n, k), k=0..n-1), n=1..14);
MATHEMATICA
b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i<1 || v<1 || n<v, 0, If[n == v, 1, Sum[Binomial[A[i, Min[i-1, h]]+j-1, j]*b[n-i*j, i-1, h, v-j], {j, 0, Min[n/i, v]}]]]]; A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, 1, Min[k, n-1]}]]; T[n_, k_] := b[n-1, n-1, k, k]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 04 2014
STATUS
approved