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A256068 Number T(n,k) of rooted identity 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 A256068 #18 Jan 04 2021 16:54:06
%S A256068 1,0,1,0,1,3,0,2,14,16,0,3,60,174,125,0,6,254,1434,2464,1296,0,12,
%T A256068 1087,10746,33362,40455,16807,0,25,4742,77556,388312,816535,763104,
%U A256068 262144,0,52,21020,551460,4191916,13617210,21501684,16328620,4782969
%N A256068 Number T(n,k) of rooted identity 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 A256068 Alois P. Heinz, <a href="/A256068/b256068.txt">Rows n = 1..141, flattened</a>
%F A256068 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A255517(n).
%e A256068 T(4,2) = 14:
%e A256068 :   0   0   0   0   0   0     0       0
%e A256068 :   |   |   |   |   |   |     |       |
%e A256068 :   1   1   2   2   2   1     1       2
%e A256068 :   |   |   |   |   |   |    / \     / \
%e A256068 :   1   2   1   2   1   2   1   2   1   2
%e A256068 :   |   |   |   |   |   |
%e A256068 :   2   1   1   1   2   1
%e A256068 :
%e A256068 :     0      0      0      0      0      0
%e A256068 :    / \    / \    / \    / \    / \    / \
%e A256068 :   1   1  2   1  1   2  2   2  1   2  2   1
%e A256068 :   |      |      |      |      |      |
%e A256068 :   2      1      1      1      2      2
%e A256068 Triangle T(n,k) begins:
%e A256068   1;
%e A256068   0,  1;
%e A256068   0,  1,    3;
%e A256068   0,  2,   14,    16;
%e A256068   0,  3,   60,   174,    125;
%e A256068   0,  6,  254,  1434,   2464,   1296;
%e A256068   0, 12, 1087, 10746,  33362,  40455,  16807;
%e A256068   0, 25, 4742, 77556, 388312, 816535, 763104, 262144;
%e A256068   ...
%p A256068 with(numtheory):
%p A256068 A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
%p A256068       k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
%p A256068     end:
%p A256068 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A256068 seq(seq(T(n, k), k=0..n-1), n=1..10);
%t A256068 A[n_, k_] := A[n, k] = If[n < 2, n, Sum[A[n - j, k] Sum[k A[d, k] d * (-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n - 1}]/(n - 1)];
%t A256068 T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
%t A256068 Table[T[n, k], {n, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, May 29 2020, after Maple *)
%Y A256068 Columns k=0-1 give: A063524 (for n>0), A004111 (for n>1):
%Y A256068 Main diagonal gives: A000272 (for n>0).
%Y A256068 Row sums give A319220(n-1).
%Y A256068 T(2n+1,n) gives A309996.
%Y A256068 Cf. A255517, A256064.
%K A256068 nonn,tabl
%O A256068 1,6
%A A256068 _Alois P. Heinz_, Mar 13 2015