A327245 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 have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 3, 0, 3, 10, 13, 0, 3, 39, 87, 75, 0, 5, 100, 510, 836, 541, 0, 11, 303, 2272, 7042, 9025, 4683, 0, 13, 782, 9999, 46628, 104255, 109110, 47293, 0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835, 0, 27, 5388, 154038, 1577256, 7676830, 19798096, 28538496, 21713032, 7087261

Offset: 0

Author

Alois P. Heinz, Sep 14 2019

Keywords

• nonn
• tabl

Examples

			T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  3,   10,    13;
  0,  3,   39,    87,     75;
  0,  5,  100,   510,    836,    541;
  0, 11,  303,  2272,   7042,   9025,    4683;
  0, 13,  782,  9999,  46628, 104255,  109110,   47293;
  0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835;
  ...
		

Links

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

Crossrefs

Columns k=0-2 give: A000007, A032020 (for n>0), A327847.

Main diagonal gives A000670.

Row sums give A321586.

T(2n,n) gives A327589.

Cf. A327244, A327588.

Programs

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)*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] 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 29 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A327588(n).