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.
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
Examples
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;
Links
- Alois P. Heinz, Rows n = 1..200, flattened
Crossrefs
Programs
-
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 *)
Comments