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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.

%I A317578 #27 Jun 13 2021 07:16:35
%S A317578 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,
%T A317578 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,
%U A317578 2,104,5,108,6,108,8,1,123,5,122,9,119,14,136,9,147,7
%N 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.
%H A317578 Alois P. Heinz, <a href="/A317578/b317578.txt">Rows n = 3..5000, flattened</a>
%F A317578 Sum_{k>=1} k * T(n,k) = A001399(n-3) = A069905(n) = A211540(n+2).
%F A317578 Sum_{k>=2} T(n,k) = A060277(n).
%F A317578 min { n >= 0 : T(n,k) > 0 } = A103277(k).
%e A317578 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].
%e A317578 T(14,2) = 2: 40 ([8,5,1], [10,2,2]) and 72 ([6,6,2], [8,3,3]).
%e A317578 T(39,3) = 1: 1200 ([20,15,4], [24,10,5], [25,8,6]).
%e A317578 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]).
%e A317578 Triangle T(n,k) begins:
%e A317578    1;
%e A317578    1;
%e A317578    2;
%e A317578    3;
%e A317578    4;
%e A317578    5;
%e A317578    7;
%e A317578    8;
%e A317578   10;
%e A317578   12;
%e A317578   12, 1;
%e A317578   12, 2;
%e A317578   19;
%e A317578   19, 1;
%e A317578   22, 1;
%p A317578 b:= proc(n) option remember; local m, c, i, j, h, w;
%p A317578       m, c:= proc() 0 end, 0; forget(m);
%p A317578       for i to iquo(n, 3) do for j from i to iquo(n-i, 2) do
%p A317578         h:= i*j*(n-j-i);
%p A317578         w:= m(h); w:= w+1; m(h):= w;
%p A317578         c:= c+x^w-x^(w-1)
%p A317578       od od; c
%p A317578     end:
%p A317578 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A317578 seq(T(n), n=3..100);
%t A317578 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 A317578 T[n_] := CoefficientList[b[n], x] // Rest;
%t A317578 T /@ Range[3, 100] // Flatten (* _Jean-François Alcover_, Jun 13 2021, after _Alois P. Heinz_ *)
%Y A317578 Cf. A001399, A060277, A069905, A103277, A211540.
%Y A317578 Row sums give A306403.
%Y A317578 Column k=1 gives A306435.
%K A317578 nonn,look,tabf
%O A317578 3,3
%A A317578 _Alois P. Heinz_, Jul 31 2018