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A327116 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 11
1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 2, 15, 27, 15, 0, 3, 32, 102, 124, 52, 0, 4, 65, 319, 656, 600, 203, 0, 5, 124, 897, 2780, 4210, 3084, 877, 0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140, 0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

Wikipedia, Partition (number theory)

FORMULA

Sum_{k=1..n} k * T(n,k) = A327557(n).

EXAMPLE

T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.

Triangle T(n,k) begins:

  1;

  0, 1;

  0, 1,   2;

  0, 2,   6,    5;

  0, 2,  15,   27,    15;

  0, 3,  32,  102,   124,     52;

  0, 4,  65,  319,   656,    600,    203;

  0, 5, 124,  897,  2780,   4210,   3084,    877;

  0, 6, 230, 2346, 10305,  23040,  27567,  16849,  4140;

  0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;

  ...

MAPLE

C:= binomial:

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))

    end:

T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):

seq(seq(T(n, k), k=0..n), n=0..12);

MATHEMATICA

c = Binomial;

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]];

T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];

Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 27 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-2 give: A000007, A000009 (for n>0), A327598.

Main diagonal gives A000110.

Row sums give A317776.

T(2n,n) gives A327556.

Cf. A255903, A326914, A326962, A327117, A327557, A309973.

Sequence in context: A325199 A185197 A323845 * A157491 A094385 A291799

Adjacent sequences:  A327113 A327114 A327115 * A327117 A327118 A327119

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 13 2019

STATUS

approved

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Last modified January 22 19:45 EST 2022. Contains 350504 sequences. (Running on oeis4.)