A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 3, 8, 6, 0, 5, 24, 42, 24, 0, 7, 60, 198, 264, 120, 0, 11, 144, 780, 1848, 1920, 720, 0, 15, 320, 2778, 10512, 18840, 15840, 5040, 0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320, 0, 30, 1486, 30186, 250128, 999720, 2129040, 2479680, 1491840, 362880

Offset

0,5

Links

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

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i).

T(n,k) = k! * A256130(n,k).

Example

T(3,1) = 3: 1a1a1a, 2a1a, 1a.
T(3,2) = 8: 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a.
T(3,3) = 6: 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   8,    6;
  0,  5,  24,   42,    24;
  0,  7,  60,  198,   264,    120;
  0, 11, 144,  780,  1848,   1920,    720;
  0, 15, 320, 2778, 10512,  18840,  15840,   5040;
  0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320;
  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    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..10);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; 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, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

Crossrefs

Columns k=0-1 give: A000007, A000041 (for n>0).

Main diagonal gives A000142.

Row sums give A278644.

Cf. A246935, A256130, A319600.

Sequence in context: A255903 A118262 A065484 * A336978 A011137 A143396

Adjacent sequences: A255967 A255968 A255969 * A255971 A255972 A255973

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 12 2015

Status

approved