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A326500 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 colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 5
1, 0, 1, 0, 2, 2, 0, 3, 8, 5, 0, 5, 22, 30, 13, 0, 7, 54, 129, 124, 42, 0, 11, 118, 428, 696, 525, 150, 0, 15, 248, 1293, 3108, 3830, 2358, 576, 0, 22, 490, 3483, 11595, 20720, 20535, 10661, 2266, 0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Rows n = 1..140, flattened

Wikipedia, Partition (number theory)

FORMULA

Sum_{k=1..n} k * T(n,k) = A326656(n).

EXAMPLE

T(3,1) = 3: 3aaa, 2aa1a, 111aaa.

T(3,2) = 8: 3aab, 3abb, 2aa1b, 2ab1b, 2ab1a, 2bb1a, 111aab, 111abb.

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

Triangle T(n,k) begins:

  1;

  0,  1;

  0,  2,   2;

  0,  3,   8,    5;

  0,  5,  22,   30,    13;

  0,  7,  54,  129,   124,    42;

  0, 11, 118,  428,   696,   525,    150;

  0, 15, 248, 1293,  3108,  3830,   2358,    576;

  0, 22, 490, 3483, 11595, 20720,  20535,  10661,  2266;

  0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966;

  ...

MAPLE

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-1, 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=0..n), n=0..12);

CROSSREFS

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

Main diagonal gives A178682.

Row sums give A326654.

T(2n,n) gives A328158.

Cf. A326656.

Sequence in context: A061314 A193383 A218033 * A255903 A118262 A065484

Adjacent sequences:  A326497 A326498 A326499 * A326501 A326502 A326503

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 12 2019

STATUS

approved

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Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)