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%I #16 May 11 2019 10:11:12
%S 0,0,1,0,1,0,0,1,1,0,0,1,1,1,0,0,1,1,2,2,0,0,1,1,2,3,3,0,0,1,1,2,4,7,
%T 6,0,0,1,1,2,4,8,15,12,0,0,1,1,2,4,9,18,34,25,0,0,1,1,2,4,9,19,43,79,
%U 52,0,0,1,1,2,4,9,20,46,102,190,113,0,0,1,1,2,4,9,20,47,110,250,459,247,0
%N Number A(n,k) of rooted trees with n nodes such that no more than k isomorphic subtrees extend from the same node; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A318757/b318757.txt">Antidiagonals n = 0..200, flattened</a>
%F A(n,k) = Sum_{j=0..k} A318758(n,j) for n > 0.
%F A(n,n+j) = A000081(n) for j >= -1.
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 2, 2, 2, 2, 2, 2, ...
%e 0, 2, 3, 4, 4, 4, 4, 4, 4, ...
%e 0, 3, 7, 8, 9, 9, 9, 9, 9, ...
%e 0, 6, 15, 18, 19, 20, 20, 20, 20, ...
%e 0, 12, 34, 43, 46, 47, 48, 48, 48, ...
%e 0, 25, 79, 102, 110, 113, 114, 115, 115, ...
%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 seq(seq(A(n, d-n), n=0..d), d=0..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 Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 11 2019, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A063524, A004111, A248869, A318850, A318851, A318852, A318853, A318854, A318855, A318856, A318857.
%Y Main diagonal gives A000081.
%Y Cf. A318753, A318758.
%K nonn,tabl
%O 0,19
%A _Alois P. Heinz_, Sep 02 2018
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