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%I #18 May 11 2019 10:11:19
%S 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,
%T 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,
%U 443,167,61,22,8,3,1,1,0,548,2285,1209,461,168,60,22,8,3,1,1
%N 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.
%C 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.
%H Alois P. Heinz, <a href="/A318758/b318758.txt">Rows n = 1..200, flattened</a>
%F T(n,k) = A318757(n,k) - A318757(n,k-1) for k > 0, A(n,0) = A063524(n).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 1, 1;
%e 0, 3, 4, 1, 1;
%e 0, 6, 9, 3, 1, 1;
%e 0, 12, 22, 9, 3, 1, 1;
%e 0, 25, 54, 23, 8, 3, 1, 1;
%e 0, 52, 138, 60, 23, 8, 3, 1, 1;
%e 0, 113, 346, 164, 61, 22, 8, 3, 1, 1;
%p h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),
%p `if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))
%p end:
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))
%p end:
%p A:= (n, k)-> `if`(n<2, n, b(n-1$2, k)):
%p T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
%p seq(seq(T(n, k), k=0..n-1), n=1..14);
%t 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]}]]];
%t 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}]]];
%t A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];
%t T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, May 11 2019, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A063524, A004111 (for n>1), A318859, A318860, A318861, A318862, A318863, A318864, A318865, A318866, A318867.
%Y Row sums give A000081.
%Y T(2n+2,n+1) gives A255705.
%Y Cf. A318754, A318757.
%K nonn,tabl
%O 1,8
%A _Alois P. Heinz_, Sep 02 2018
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