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

0,5

Links

Alois P. Heinz, Rows n = 1..140, flattened

Wikipedia, Partition (number theory)

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A366730 A193383 A218033 * A255903 A118262 A065484

Adjacent sequences: A326497 A326498 A326499 * A326501 A326502 A326503

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 12 2019

Status

approved