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A245151 Number T(n,k) of n-node unlabeled rooted trees with thickening limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows. 12
1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 5, 1, 0, 0, 1, 0, 7, 3, 1, 0, 0, 1, 0, 12, 3, 1, 0, 0, 0, 1, 0, 17, 8, 1, 1, 0, 0, 0, 1, 0, 28, 9, 3, 1, 0, 0, 0, 0, 1, 0, 42, 21, 3, 1, 1, 0, 0, 0, 0, 1, 0, 69, 28, 5, 1, 1, 0, 0, 0, 0, 0, 1, 0, 105, 56, 9, 3, 1, 1, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
In a rooted tree with thickening limbs the outdegree of a parent node is smaller than or equal to the outdegree of any of its non-leaf child nodes.
T(n+1,1) = Sum_{k=0..n-1} T(n,k) for n>=1.
T(n+1,n) = T(2n+1,n) = 1 for n>=0.
T(n,1+floor((n-1)/2)) = 0 for n>3.
LINKS
EXAMPLE
The A245152(5) = 5 5-node rooted trees with thickening 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 : : :
: : : :
: ------1------ : ---2--- : ---4--- :
Thus row 5 = [0, 3, 1, 0, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 0, 1;
0, 3, 1, 0, 1;
0, 5, 1, 0, 0, 1;
0, 7, 3, 1, 0, 0, 1;
0, 12, 3, 1, 0, 0, 0, 1;
0, 17, 8, 1, 1, 0, 0, 0, 1;
0, 28, 9, 3, 1, 0, 0, 0, 0, 1;
0, 42, 21, 3, 1, 1, 0, 0, 0, 0, 1;
0, 69, 28, 5, 1, 1, 0, 0, 0, 0, 0, 1;
0, 105, 56, 9, 3, 1, 1, 0, 0, 0, 0, 0, 1;
0, 176, 81, 12, 3, 1, 1, 0, 0, 0, 0, 0, 0, 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, 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=k..n-1))
end:
T:= (n, k)-> b(n-1$2, k$2):
seq(seq(T(n, k), k=0..n-1), n=1..20);
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, 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, k, n-1}]]; T[n_, k_] := b[n-1, n-1, k, k]; Table[ Table[T[n, k], {k, 0, n - 1}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007(n-1), A245152(n-1), A245142, A245143, A245144, A245145, A245146, A245147, A245148, A245149, A245150.
Row sums give A245152.
Cf. A244657 (thinning limbs).
Sequence in context: A318371 A363157 A363952 * A243978 A356898 A106844
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 12 2014
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)