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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.
7
1, 0, 1, 0, 2, 3, 0, 4, 18, 16, 0, 9, 89, 201, 125, 0, 20, 418, 1830, 2720, 1296, 0, 48, 1962, 14845, 39720, 43580, 16807, 0, 115, 9268, 114624, 492276, 934455, 809760, 262144, 0, 286, 44375, 866148, 5613775, 16413510, 23991063, 17152163, 4782969
OFFSET
1,5
LINKS
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A242249(n,k-i).
EXAMPLE
T(3,2) = 3:
o o o
| | / \
1 2 1 2
| |
2 1
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 3;
0, 4, 18, 16;
0, 9, 89, 201, 125;
0, 20, 418, 1830, 2720, 1296;
0, 48, 1962, 14845, 39720, 43580, 16807;
0, 115, 9268, 114624, 492276, 934455, 809760, 262144;
...
MAPLE
with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n-1), n=1..10);
MATHEMATICA
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[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, from Maple *)
CROSSREFS
Columns k=0-1 give: A063524 (for n>0), A000081 (for n>1):
Main diagonal gives: A000272 (for n>0).
T(2n+1,n) gives A309994.
Sequence in context: A140326 A261781 A211402 * A126436 A102394 A145091
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 13 2015
STATUS
approved