A194703 - OEIS

%I #29 Sep 02 2023 20:32:56

%S 3,2,1,0,1,2,1,0,1,1,0,1,0,1,1,0,0,1,0,1,1,3,2,1,0,1,2,1,0,1,1,0,1,0,

%T 1,1,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,1,1,0,0,

%U 0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0

%N Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (3 + m).

%C Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 3. For further information see A182703 and A135010.

%F T(k,m) = A182703(3+m,k), with T(k,m) = 0 if k > 3+m.

%F T(k,m) = A194812(3+m,k).

%e Triangle begins:

%e 3,

%e 2, 1,

%e 0, 1, 2,

%e 1, 0, 1, 1,

%e 0, 1, 0, 1, 1,

%e 0, 0, 1, 0, 1, 1,

%e 0, 0, 0, 1, 0, 1, 1,

%e 0, 0, 0, 0, 1, 0, 1, 1,

%e 0, 0, 0, 0, 0, 1, 0, 1, 1,

%e 0, 0, 0, 0, 0, 0, 1, 0, 1, 1,

%e ...

%e For k = 1 and m = 1, T(1,1) = 3 because there are three parts of size 1 in the last section of the set of partitions of 4, since 3 + m = 4, so a(1) = 3.

%e For k = 2 and m = 1, T(2,1) = 2 because there are two parts of size 2 in the last section of the set of partitions of 4, since 3 + m = 4, so a(2) = 2.

%Y Always the sum of row k = p(3) = A000041(3) = 3.

%Y The first (0-10) members of this family of triangles are A023531, A129186, A194702, this sequence, A194704-A194710.

%Y Cf. A135010, A138121, A182712-A182714, A194812.

%K nonn,tabl

%O 1,1

%A _Omar E. Pol_, Feb 05 2012