A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1

Offset: 0

Author

Alois P. Heinz, Aug 29 2019

Keywords

• nonn
• tabl

Comments

For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

T is the convolution triangle of A000009 (see A357368). - Peter Luschny, Oct 19 2022

Examples

			T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  3,  8,   9,   4,   1;
  0,  4, 14,  19,  14,   5,   1;
  0,  5, 22,  39,  36,  20,   6,   1;
  0,  6, 34,  72,  85,  60,  27,   7,  1;
  0,  8, 50, 128, 180, 160,  92,  35,  8, 1;
  0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Partition (number theory)

Crossrefs

Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.

Main diagonal and lower diagonals give: A000012, A001477, A000096.

Row sums give A304969.

T(2n,n) gives A324595.

Cf. A060642, A286335, A325915.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, 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);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
          (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,    b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).

Sum_{k=1..n} k * T(n,k) = A325915(n).

G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021