A327244 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 2, 0, 3, 10, 8, 0, 3, 27, 54, 31, 0, 5, 70, 255, 336, 147, 0, 11, 223, 1222, 2692, 2580, 899, 0, 13, 508, 4467, 15512, 25330, 19566, 5777, 0, 19, 1193, 15540, 78819, 194075, 248976, 160377, 41024, 0, 27, 2822, 52981, 375440, 1303250, 2463534, 2593339, 1430288, 322488

Offset

0,6

Links

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

Formula

Sum_{k=1..n} k * T(n,k) = A327595(n).

Example

T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
T(3,3) = 8: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    2;
  0,  3,   10,     8;
  0,  3,   27,    54,    31;
  0,  5,   70,   255,   336,    147;
  0, 11,  223,  1222,  2692,   2580,    899;
  0, 13,  508,  4467, 15512,  25330,  19566,   5777;
  0, 19, 1193, 15540, 78819, 194075, 248976, 160377, 41024;
  ...

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give: A000007, A032020 (for n>0), A327841.

Main diagonal gives A120774.

Row sums give A309670.

T(2n,n) gives A327596.

Cf. A327245, A327595.

Sequence in context: A236396 A251592 A365985 * A325816 A121065 A077928

Adjacent sequences: A327241 A327242 A327243 * A327245 A327246 A327247

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 14 2019

Status

approved