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Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (2 + m).
9

%I #29 Nov 30 2013 21:30:34

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

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

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

%N Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (2 + m).

%C Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 2. For further information see A182703 and A135010.

%F T(k,m) = A182703(2+m,k), with T(k,m) = 0 if k > 2+m.

%F T(k,m) = A194812(2+m,k).

%e Triangle begins:

%e 2,

%e 0, 2,

%e 1, 0, 1,

%e 0, 1, 0, 1,

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

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

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

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

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

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

%e ...

%e For k = 1 and m = 1; T(1,1) = 2 because there are two parts of size 1 in the last section of the set of partitions of 3, since 2 + m = 3, so a(1) = 2. For k = 2 and m = 1; T(2,1) = 0 because there are no parts of size 2 in the last section of the set of partitions of 3, since 2 + m = 3, so a(2) = 0.

%Y Always the sum of row k = p(2) = A000041(n) = 2.

%Y The first (0-10) members of this family of triangles are A023531, A129186, this sequence, A194703-A194710.

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

%K nonn,tabl

%O 1,1

%A _Omar E. Pol_, Feb 05 2012