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.

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

Offset: 3

Author

Alois P. Heinz, Jul 31 2018

Keywords

• nonn
• look
• tabf

Examples

			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;
		

Links

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

Crossrefs

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

Row sums give A306403.

Column k=1 gives A306435.

Programs

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);

Mathematica

b[n_] := b[n] = Module[{m, c, i, j, h, w} , m[_] = 0; c = 0; For[i = 1, i <= Quotient[n, 3], i++, For[j = i, j <= Quotient[n - i, 2], j++, h = i*j*(n-j-i); w = m[h]; w++; m[h] = w; c = c + x^w - x^(w-1)]]; c];
T[n_] := CoefficientList[b[n], x] // Rest;
T /@ Range[3, 100] // Flatten (* Jean-François Alcover, Jun 13 2021, after Alois P. Heinz *)

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).