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

0,6

Links

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

Formula

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

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 *)

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.

Sequence in context: A177785 A212036 A191619 * A157525 A157521 A176005

Adjacent sequences: A327242 A327243 A327244 * A327246 A327247 A327248

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 14 2019

Status

approved