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.

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

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition (number theory)

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.

Cf. A001787, A255903, A326914, A326962, A327116, A327118.

Sequence in context: A247126 A342134 A349740 * A359107 A229223 A128749

Adjacent sequences: A327114 A327115 A327116 * A327118 A327119 A327120

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 13 2019

Status

approved