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A255903
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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.
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33
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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
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OFFSET
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0,5
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COMMENTS
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T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.
In the case of exactly one color (k=1) each multiset of monochrome objects is fully described by its size and a collection of sizes corresponds to an integer partition. In the case of distinct colors for all objects (k=n) every multiset collection is a set partition.
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LINKS
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Alois P. Heinz, Rows n = 0..140, flattened
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A075196(n,k-i).
Sum_{k=0..n} k * T(n,k) = A317178(n).
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EXAMPLE
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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;
...
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MAPLE
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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);
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MATHEMATICA
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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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CROSSREFS
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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.
Cf. A075196, A317178, A326914, A326962, A327116, A327117.
Sequence in context: A193383 A218033 A326500 * A118262 A065484 A255970
Adjacent sequences: A255900 A255901 A255902 * A255904 A255905 A255906
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Mar 10 2015
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STATUS
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approved
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