A326500 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 3, 8, 5, 0, 5, 22, 30, 13, 0, 7, 54, 129, 124, 42, 0, 11, 118, 428, 696, 525, 150, 0, 15, 248, 1293, 3108, 3830, 2358, 576, 0, 22, 490, 3483, 11595, 20720, 20535, 10661, 2266, 0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966

Offset: 0

Author

Alois P. Heinz, Sep 12 2019

Keywords

• nonn
• tabl

Examples

			T(3,1) = 3: 3aaa, 2aa1a, 111aaa.
T(3,2) = 8: 3aab, 3abb, 2aa1b, 2ab1b, 2ab1a, 2bb1a, 111aab, 111abb.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   8,    5;
  0,  5,  22,   30,    13;
  0,  7,  54,  129,   124,    42;
  0, 11, 118,  428,   696,   525,    150;
  0, 15, 248, 1293,  3108,  3830,   2358,    576;
  0, 22, 490, 3483, 11595, 20720,  20535,  10661,  2266;
  0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..140, flattened
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-1 give: A000007, A000041 (for n>0).

Main diagonal gives A178682.

Row sums give A326654.

T(2n,n) gives A328158.

Cf. A326656.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^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[Function[t, b[n-t, Min[n-t, i-1], k] Binomial[k + t - 1, t]][i j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A326656(n).