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A326962 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 distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns. 9
1, 1, 2, 2, 1, 5, 12, 18, 20, 18, 15, 11, 6, 3, 1, 15, 64, 166, 332, 566, 864, 1214, 1596, 1975, 2320, 2600, 2780, 2842, 2780, 2600, 2320, 1979, 1608, 1238, 908, 626, 404, 246, 136, 69, 32, 12, 4, 1, 52, 340, 1315, 3895, 9770, 21848, 44880, 86275, 157140 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.

LINKS

Alois P. Heinz, Columns k = 0..10, flattened

Wikipedia, Partition (number theory)

FORMULA

Sum_{k=1..n} k * T(n,k) = A327115(n).

T(n*2^(n-1),n) = T(A001787(n),n) = 1.

T(n*2^(n-1)-1,n) = n for n >= 2.

EXAMPLE

T(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.

Triangle T(n,k) begins:

  1;

     1;

        2;

        2,  5;

        1, 12,   15;

           18,   64,    52;

           20,  166,   340,    203;

           18,  332,  1315,   1866,    877;

           15,  566,  3895,   9930,  10710,   4140;

           11,  864,  9770,  39960,  74438,  64520,  21147;

            6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;

  ...

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), 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), n=k..k*2^(k-1)), k=0..5);

CROSSREFS

Main diagonal gives A000110.

Row sums give A116539.

Column sums give A003465.

Cf. A001787, A255903, A326914 (this triangle read by rows), A327115, A327116, A327117.

Sequence in context: A297628 A329429 A326617 * A280245 A290013 A220963

Adjacent sequences:  A326959 A326960 A326961 * A326963 A326964 A326965

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 13 2019

STATUS

approved

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Last modified February 19 00:54 EST 2020. Contains 332028 sequences. (Running on oeis4.)