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A245120
Number T(n,k) of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=max-index-of-row(n), read by rows.
12
1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, 1, 0, 1, 8, 2, 0, 1, 12, 4, 0, 1, 22, 9, 0, 1, 36, 17, 2, 0, 1, 63, 35, 3, 0, 1, 107, 67, 9, 0, 1, 188, 131, 20, 0, 1, 327, 249, 46, 1, 0, 1, 578, 484, 94, 4, 0, 1, 1020, 922, 202, 11, 0, 1, 1820, 1775, 412, 28
OFFSET
1,14
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
EXAMPLE
The A124346(7) = 6 7-node rooted identity 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 : o o o o : o :
: | : | | | : :
: o : o o o : :
: | : | : :
: o : o : :
: | : : :
: o : : :
: : : :
: -1- : -------------2------------ : --3-- :
Thus row 7 = [0, 1, 4, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1;
0, 1, 1;
0, 1, 1;
0, 1, 3;
0, 1, 4, 1;
0, 1, 8, 2;
0, 1, 12, 4;
0, 1, 22, 9;
0, 1, 36, 17, 2;
0, 1, 63, 35, 3;
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, add(binomial(A(i, min(i-1, h)), 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:
g:= proc(n) local k; if n=1 then 0 else
for k while T(n, k)>0 do od; k-1 fi
end:
T:= (n, k)-> b(n-1$2, k$2):
seq(seq(T(n, k), k=0..g(n)), n=1..25);
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, Sum[Binomial[A[i, Min[i-1, h]], 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]}]]; g[n_] := If[n==1, 0, For[k=1, T[n, k]>0, k++]; k-1]; T[n_, k_] := b[n-1, n-1, k, k]; Table[T[n, k], {n, 1, 25}, {k, 0, g[n]}] // Flatten (* Jean-François Alcover, Jan 18 2017, translated from Maple *)
CROSSREFS
Column k=0-10 give: A000007(n-1), A000012 (for n>1), A245121, A245122, A245123, A245124, A245125, A245126, A245127, A245128, A245129.
Row sums give A124346.
Cf. A244657.
Sequence in context: A247629 A178116 A238709 * A226912 A177330 A197126
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jul 12 2014
STATUS
approved