A317578 - OEIS

%I #27 Jun 13 2021 07:16:35

%S 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 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 2,104,5,108,6,108,8,1,123,5,122,9,119,14,136,9,147,7

%N 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 Alois P. Heinz, <a href="/A317578/b317578.txt">Rows n = 3..5000, flattened</a>

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

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

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

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

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

%e 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 Triangle T(n,k) begins:

%e    1;

%e    1;

%e    2;

%e    3;

%e    4;

%e    5;

%e    7;

%e    8;

%e   10;

%e   12;

%e   12, 1;

%e   12, 2;

%e   19;

%e   19, 1;

%e   22, 1;

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

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

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

%p         h:= i*j*(n-j-i);

%p         w:= m(h); w:= w+1; m(h):= w;

%p         c:= c+x^w-x^(w-1)

%p       od od; c

%p     end:

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

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

%t 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 T[n_] := CoefficientList[b[n], x] // Rest;

%t T /@ Range[3, 100] // Flatten (* _Jean-François Alcover_, Jun 13 2021, after _Alois P. Heinz_ *)

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

%Y Row sums give A306403.

%Y Column k=1 gives A306435.

%K nonn,look,tabf

%O 3,3

%A _Alois P. Heinz_, Jul 31 2018