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A317578 Number T(n,k) of distinct integers that are product of the parts of exactly k partitions of n into 3 positive parts; triangle T(n,k), n>=3, k>=1, read by rows. 6
1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 12, 1, 12, 2, 19, 19, 1, 22, 1, 27, 28, 1, 31, 1, 31, 3, 38, 1, 42, 1, 46, 1, 50, 1, 50, 3, 57, 2, 51, 7, 64, 3, 71, 2, 70, 5, 77, 4, 85, 3, 86, 5, 84, 9, 104, 2, 104, 5, 108, 6, 108, 8, 1, 123, 5, 122, 9, 119, 14, 136, 9, 147, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

LINKS

Alois P. Heinz, Rows n = 3..5000, flattened

FORMULA

Sum_{k>=1} k * T(n,k) = A001399(n-3) = A069905(n) = A211540(n+2).

Sum_{k>=2} T(n,k) = A060277(n).

min { n >= 0 : T(n,k) > 0 } = A103277(k).

EXAMPLE

T(13,2) = 1: only 36 is product of the parts of exactly 2 partitions of 13 into 3 positive parts: [6,6,1], [9,2,2].

T(14,2) = 2: 40 ([8,5,1], [10,2,2]) and 72 ([6,6,2], [8,3,3]).

T(39,3) = 1: 1200 ([20,15,4], [24,10,5], [25,8,6]).

T(49,3) = 2: 3024 ([24,18,7], [27,14,8], [28,12,9]) and 3600 ([20,20,9], [24,15,10], [25,12,12]).

Triangle T(n,k) begins:

   1;

   1;

   2;

   3;

   4;

   5;

   7;

   8;

  10;

  12;

  12, 1;

  12, 2;

  19;

  19, 1;

  22, 1;

MAPLE

b:= proc(n) option remember; local m, c, i, j, h, w;

      m, c:= proc() 0 end, 0; forget(m);

      for i to iquo(n, 3) do for j from i to iquo(n-i, 2) do

        h:= i*j*(n-j-i);

        w:= m(h); w:= w+1; m(h):= w;

        c:= c+x^w-x^(w-1)

      od od; c

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):

seq(T(n), n=3..100);

CROSSREFS

Cf. A001399, A060277, A069905, A103277, A211540.

Row sums give A306403.

Column k=1 gives A306435.

Sequence in context: A305414 A317296 A103740 * A306435 A034155 A306403

Adjacent sequences:  A317575 A317576 A317577 * A317579 A317580 A317581

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Jul 31 2018

STATUS

approved

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Last modified May 6 13:04 EDT 2021. Contains 343585 sequences. (Running on oeis4.)