A213238 - OEIS

%I #15 Feb 19 2021 11:17:06

%S 1,3,8,12,18,24,32,40,42,50,60,64,72,84,88,90,98,99,105,112,120,128,

%T 130,135,144,162,168,180,182,192,200,208,210,220,231,242,252,264,266,

%U 272,280,288,294,300,312,315,320,324,330,338,340,360,364,378,392,400

%N Triangle T(n,k) in which n-th row lists in increasing order the distinct values v satisfying v = sum of elements in S = product of elements in P for a partition of {1,...,n} into two sets S and P.

%H Alois P. Heinz, <a href="/A213238/b213238.txt">Rows n = 1..798, flattened</a>

%F T(n,1) = floor((n-1)^2/2) = A007590(n-1) for n>=5.

%e For n=1 v=1 satisfies the condition with S={1}, P={} => row 1 = [1].

%e For n=2 no v can be found => row 2 is empty: [].

%e For n=3 there is one solution: S={1,2}, P={3}, v=3 => row 3 = [3].

%e For n=10 we have three partitions of {1,2,...,10} into S and P satisfying v = Sum_{i:S} i = Product_{k:P} k but there are only two distinct values v: S={2,3,5,6,7,8,9}, P={1,4,10}, v=40; S={4,5,6,8,9,10}, P={1,2,3,7}, v=42; S={1,2,3,4,5,8,9,10}, P={6,7}, v=42 => row 10 = [40, 42].

%e Triangle T begins:

%e    1;

%e     ;

%e    3;

%e     ;

%e    8;

%e   12;

%e   18;

%e   24;

%e   32;

%e   40, 42;

%e   50;

%e   60, 64;

%e   72;

%e   84, 88, 90;

%e   ...

%p b:= proc(n, s, p)

%p       `if`(s=p, {s}, `if`(n<1, {}, {b(n-1, s, p)[],

%p       `if`(s-n<p*n, {}, b(n-1, s-n, p*n))[]}))

%p     end:

%p T:= n-> sort([b(n, n*(n+1)/2, 1)[]])[]:

%p seq(T(n), n=1..30);

%t b[n_, s_, p_] :=

%t      If[s == p, {s}, If[n < 1, {}, {b[n-1, s, p],

%t      If[s-n < p*n, {}, b[n-1, s-n, p*n]]} // Union]];

%t T[n_] := Sort[b[n, n(n+1)/2, 1] // Flatten] // Union;

%t Array[T, 30] // Flatten (* _Jean-François Alcover_, Feb 19 2021, after _Alois P. Heinz_ *)

%Y Row lengths (or number of solutions) are in A213237.

%Y T(n,1) = A007590(n-1) for n>=5.

%K nonn,tabf

%O 1,2

%A _Alois P. Heinz_, Jun 07 2012