A255903 Number T(n,k) of collections of nonempty multisets with a total of n objects of exactly k colors; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 2, 2, 0, 3, 8, 5, 0, 5, 23, 33, 15, 0, 7, 56, 141, 144, 52, 0, 11, 127, 492, 848, 675, 203, 0, 15, 268, 1518, 3936, 5190, 3396, 877, 0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140, 0, 30, 1072, 11567, 57420, 154410, 240012, 216006, 104656, 21147
Offset: 0
Examples
T(3,1) = 3: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}.
T(3,2) = 8: {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}.
T(3,3) = 5: {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 2;
0, 3, 8, 5;
0, 5, 23, 33, 15;
0, 7, 56, 141, 144, 52;
0, 11, 127, 492, 848, 675, 203;
0, 15, 268, 1518, 3936, 5190, 3396, 877;
0, 22, 547, 4320, 15800, 30710, 32835, 18270, 4140;
...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Columns k=0-10 give: A000007, A000041 (for n>0), A255942, A255943, A255944, A255945, A255946, A255947, A255948, A255949, A255950.
Main and lower diagonals give: A000110, A255951, A255952, A255953, A255954, A255955, A255956, A255957, A255958, A255959, A255960.
Row sums give A255906.
Antidiagonal sums give A258450.
T(2n,n) gives A255907.
Programs
-
Maple
with(numtheory): A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)* add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n) 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), n=0..12); -
Mathematica
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i * Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12} ] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)
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