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A327117
Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
9
1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 18, 15, 0, 1, 10, 45, 84, 52, 0, 1, 14, 94, 298, 415, 203, 0, 1, 18, 174, 844, 1995, 2178, 877, 0, 1, 23, 300, 2081, 7440, 13638, 12131, 4140, 0, 1, 28, 486, 4652, 23670, 64898, 95823, 71536, 21147, 0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975
OFFSET
0,6
COMMENTS
The sequence of column k satisfies a linear recurrence with constant coefficients of order k*2^(k-1) = A001787(k).
LINKS
FORMULA
Sum_{k=1..n} k * T(n,k) = A327118(n).
EXAMPLE
T(3,2) = 4: 2ab1a, 2ab1b, 1a1a1b, 1a1b1b.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 4, 5;
0, 1, 7, 18, 15;
0, 1, 10, 45, 84, 52;
0, 1, 14, 94, 298, 415, 203;
0, 1, 18, 174, 844, 1995, 2178, 877;
0, 1, 23, 300, 2081, 7440, 13638, 12131, 4140;
0, 1, 28, 486, 4652, 23670, 64898, 95823, 71536, 21147;
0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975;
...
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)+j-1, 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
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] + j - 1, 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, Oct 04 2019, after Alois P. Heinz *)
CROSSREFS
Columns k=0-3 give: A000007, A057427, A014616(n-1) for n>1, A327842.
Main diagonal gives A000110.
Row sums give A116540.
T(2n,n) gives A327843.
Sequence in context: A381567 A342134 A349740 * A359107 A229223 A128749
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 13 2019
STATUS
approved