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A327116
Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
11
1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 2, 15, 27, 15, 0, 3, 32, 102, 124, 52, 0, 4, 65, 319, 656, 600, 203, 0, 5, 124, 897, 2780, 4210, 3084, 877, 0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140, 0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147
OFFSET
0,6
LINKS
FORMULA
Sum_{k=1..n} k * T(n,k) = A327557(n).
EXAMPLE
T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 2, 6, 5;
0, 2, 15, 27, 15;
0, 3, 32, 102, 124, 52;
0, 4, 65, 319, 656, 600, 203;
0, 5, 124, 897, 2780, 4210, 3084, 877;
0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140;
0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;
...
MAPLE
C:= binomial:
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)*C(C(k+i-1, i), j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
c = Binomial;
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] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)
CROSSREFS
Columns k=0-2 give: A000007, A000009 (for n>0), A327598.
Main diagonal gives A000110.
Row sums give A317776.
T(2n,n) gives A327556.
Sequence in context: A136426 A325199 A185197 * A157491 A094385 A355260
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 13 2019
STATUS
approved