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A309973
Number T(n,k) of colored integer partitions of n using all colors of a k-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
7
1, 0, 1, 0, 1, 3, 0, 2, 6, 10, 0, 2, 21, 42, 47, 0, 3, 42, 177, 264, 246, 0, 4, 90, 619, 1746, 2095, 1602, 0, 5, 176, 1809, 7556, 16085, 16608, 11481, 0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503, 0, 8, 640, 13961, 120964, 522890, 1262832, 1842659, 1503232, 871030
OFFSET
0,6
LINKS
FORMULA
Sum_{k=1..n} k * T(n,k) = A327680(n).
EXAMPLE
T(3,1) = 2: 3aaa, 2aa1a.
T(3,2) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
T(3,3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 3;
0, 2, 6, 10;
0, 2, 21, 42, 47;
0, 3, 42, 177, 264, 246;
0, 4, 90, 619, 1746, 2095, 1602;
0, 5, 176, 1809, 7556, 16085, 16608, 11481;
0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)
CROSSREFS
Columns k=0-2 give: A000007, A000009 (for n>0), A327890.
Main diagonal gives A005651.
Row sums give A327679.
T(2n,n) gives A327681.
Sequence in context: A072328 A135040 A048733 * A298058 A298707 A274417
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 21 2019
STATUS
approved