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A256064 Number T(n,k) of rooted trees with n nodes and colored non-root nodes using exactly k colors; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

%I A256064 #20 Jan 04 2021 16:52:15
%S A256064 1,0,1,0,2,3,0,4,18,16,0,9,89,201,125,0,20,418,1830,2720,1296,0,48,
%T A256064 1962,14845,39720,43580,16807,0,115,9268,114624,492276,934455,809760,
%U A256064 262144,0,286,44375,866148,5613775,16413510,23991063,17152163,4782969
%N A256064 Number T(n,k) of rooted trees with n nodes and colored non-root nodes using exactly k colors; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
%H A256064 Alois P. Heinz, <a href="/A256064/b256064.txt">Rows n = 1..141, flattened</a>
%F A256064 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A242249(n,k-i).
%e A256064 T(3,2) = 3:
%e A256064   o    o      o
%e A256064   |    |     / \
%e A256064   1    2    1   2
%e A256064   |    |
%e A256064   2    1
%e A256064 Triangle T(n,k) begins:
%e A256064   1;
%e A256064   0,   1;
%e A256064   0,   2,    3;
%e A256064   0,   4,   18,     16;
%e A256064   0,   9,   89,    201,    125;
%e A256064   0,  20,  418,   1830,   2720,   1296;
%e A256064   0,  48, 1962,  14845,  39720,  43580,  16807;
%e A256064   0, 115, 9268, 114624, 492276, 934455, 809760, 262144;
%e A256064   ...
%p A256064 with(numtheory):
%p A256064 A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
%p A256064       A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
%p A256064     end:
%p A256064 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A256064 seq(seq(T(n, k), k=0..n-1), n=1..10);
%t A256064 A[n_, k_] := A[n, k] = If[n < 2, n, (Sum[Sum[d*A[d, k], {d, Divisors[j]}]* A[n - j, k]*k, {j, 1, n - 1}])/(n - 1)];
%t A256064 T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A256064 Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Jan 07 2020, from Maple *)
%Y A256064 Columns k=0-1 give: A063524 (for n>0), A000081 (for n>1):
%Y A256064 Main diagonal gives: A000272 (for n>0).
%Y A256064 T(2n+1,n) gives A309994.
%Y A256064 Cf. A242249, A256068.
%K A256064 nonn,tabl
%O A256064 1,5
%A A256064 _Alois P. Heinz_, Mar 13 2015