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A326616 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), n>=0, A185283(n)<=k<=n, read by rows. 7
1, 1, 2, 2, 5, 1, 9, 13, 9, 44, 42, 10, 96, 225, 150, 9, 152, 680, 1098, 576, 3, 155, 1350, 4155, 5201, 2266, 124, 2180, 11730, 26642, 26904, 9966, 140, 3751, 30300, 106281, 182000, 149832, 47466, 160, 6050, 69042, 348061, 896392, 1229760, 855240, 237019 (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, Rows n = 0..200, flattened

Wikipedia, Partition (number theory)

FORMULA

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).

Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

EXAMPLE

T(3,2) = 2: 2a1b, 2b1a.

T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc

Triangle T(n,k) begins:

  1;

     1;

        2;

        2,  5;

        1,  9,  13;

            9,  44,   42;

           10,  96,  225,   150;

            9, 152,  680,  1098,    576;

            3, 155, 1350,  4155,   5201,   2266;

               124, 2180, 11730,  26642,  26904,   9966;

               140, 3751, 30300, 106281, 182000, 149832, 47466;

               ...

MAPLE

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:

h:= proc(n) option remember; local k; for k from

      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od

    end:

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

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

    end:

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

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

CROSSREFS

Main diagonal gives A178682.

Row sums give A326648.

Column sums give A326650.

Cf. A000203, A185283, A326617 (this triangle read by columns), A326649, A326651.

Sequence in context: A165922 A337293 A307834 * A249033 A068762 A326914

Adjacent sequences:  A326613 A326614 A326615 * A326617 A326618 A326619

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 12 2019

STATUS

approved

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Last modified January 28 06:59 EST 2021. Contains 340490 sequences. (Running on oeis4.)