A318859 Number of rooted trees with n nodes such that two equals the maximal number of isomorphic subtrees extending from the same node.

0, 1, 1, 4, 9, 22, 54, 138, 346, 889, 2285, 5928, 15436, 40424, 106230, 280305, 741912, 1969816, 5243942, 13995807, 37439883, 100371907, 269623436, 725638613, 1956352468, 5283171593, 14289645110, 38707131195, 104995130162, 285184002486, 775586517781

Offset: 2

Author

Alois P. Heinz, Sep 04 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..2213

Crossrefs

Column k=2 of A318758.

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)):
a:= n-> (k-> A(n, k)-A(n, k-1))(2):
seq(a(n), n=2..32);

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]];
a[n_] := A[n, 2] - A[n, 1];
Table[a[n], {n, 2, 32}] (* Jean-François Alcover, Dec 01 2023, after Alois P. Heinz *)