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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A011137 A143396 A244129

Adjacent sequences:  A255967 A255968 A255969 * A255971 A255972 A255973

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 12 2015

STATUS

approved

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Last modified September 15 04:09 EDT 2019. Contains 327062 sequences. (Running on oeis4.)