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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
OFFSET
3,3
LINKS
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
Row sums give A306403.
Column k=1 gives A306435.
Sequence in context: A373289 A317296 A103740 * A306435 A034155 A306403
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Jul 31 2018
STATUS
approved