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

3,3

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);

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 *)

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