login
A318758
Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of isomorphic subtrees extending from the same node; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
12
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 52, 138, 60, 23, 8, 3, 1, 1, 0, 113, 346, 164, 61, 22, 8, 3, 1, 1, 0, 247, 889, 443, 167, 61, 22, 8, 3, 1, 1, 0, 548, 2285, 1209, 461, 168, 60, 22, 8, 3, 1, 1
OFFSET
1,8
COMMENTS
T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k < n. T(n,k) = 0 for k >= n.
LINKS
FORMULA
T(n,k) = A318757(n,k) - A318757(n,k-1) for k > 0, A(n,0) = A063524(n).
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 1, 1;
0, 3, 4, 1, 1;
0, 6, 9, 3, 1, 1;
0, 12, 22, 9, 3, 1, 1;
0, 25, 54, 23, 8, 3, 1, 1;
0, 52, 138, 60, 23, 8, 3, 1, 1;
0, 113, 346, 164, 61, 22, 8, 3, 1, 1;
MAPLE
h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),
`if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))
end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))
end:
A:= (n, k)-> `if`(n<2, n, b(n-1$2, k)):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
MATHEMATICA
h[n_, m_, t_, k_] := h[n, m, t, k] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1, k], {j, 1, Min[k, m]}]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*h[A[i, k], j, 0, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 11 2019, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A063524, A004111 (for n>1), A318859, A318860, A318861, A318862, A318863, A318864, A318865, A318866, A318867.
Row sums give A000081.
T(2n+2,n+1) gives A255705.
Sequence in context: A124816 A238349 A318754 * A334192 A124790 A325734
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 02 2018
STATUS
approved