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A318757 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.

%I A318757 #16 May 11 2019 10:11:12
%S A318757 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 A318757 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 A318757 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 A318757 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 A318757 Alois P. Heinz, <a href="/A318757/b318757.txt">Antidiagonals n = 0..200, flattened</a>
%F A318757 A(n,k) = Sum_{j=0..k} A318758(n,j) for n > 0.
%F A318757 A(n,n+j) = A000081(n) for j >= -1.
%e A318757 Square array A(n,k) begins:
%e A318757   0,  0,  0,   0,   0,   0,   0,   0,   0, ...
%e A318757   1,  1,  1,   1,   1,   1,   1,   1,   1, ...
%e A318757   0,  1,  1,   1,   1,   1,   1,   1,   1, ...
%e A318757   0,  1,  2,   2,   2,   2,   2,   2,   2, ...
%e A318757   0,  2,  3,   4,   4,   4,   4,   4,   4, ...
%e A318757   0,  3,  7,   8,   9,   9,   9,   9,   9, ...
%e A318757   0,  6, 15,  18,  19,  20,  20,  20,  20, ...
%e A318757   0, 12, 34,  43,  46,  47,  48,  48,  48, ...
%e A318757   0, 25, 79, 102, 110, 113, 114, 115, 115, ...
%p A318757 h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),
%p A318757       `if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))
%p A318757     end:
%p A318757 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A318757       add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))
%p A318757     end:
%p A318757 A:= (n, k)-> `if`(n<2, n, b(n-1$2, k)):
%p A318757 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A318757 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 A318757 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 A318757 A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];
%t A318757 Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 11 2019, after _Alois P. Heinz_ *)
%Y A318757 Columns k=0-10 give: A063524, A004111, A248869, A318850, A318851, A318852, A318853, A318854, A318855, A318856, A318857.
%Y A318757 Main diagonal gives A000081.
%Y A318757 Cf. A318753, A318758.
%K A318757 nonn,tabl
%O A318757 0,19
%A A318757 _Alois P. Heinz_, Sep 02 2018