%I #19 Oct 23 2024 14:40:18

%S 1,1,0,2,1,0,3,0,0,0,5,2,1,0,0,7,1,0,0,0,0,11,4,1,1,0,0,0,15,3,2,0,0,

%T 0,0,0,22,8,2,1,1,0,0,0,0,30,7,3,1,0,0,0,0,0,0,42,15,6,3,1,1,0,0,0,0,

%U 0,56,15,6,2,1,0,0,0,0,0,0,0,77,27,10

%N Square array read by antidiagonals: T(n,k) = number of parts of size k in the last section of the set of partitions of n.

%C It appears that in the column k, starting in row n, the sum of k successive terms is equal to A000041(n-1).

%F It appears that A000041(n) = Sum_{j=1..k} T(n+j,k), n >= 0, k >= 1.

%e Array begins:

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

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

%e . 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

%e . 3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,...

%e . 5, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0,...

%e . 7, 4, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0,...

%e . 11, 3, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0,...

%e . 15, 8, 3, 3, 1, 1, 0, 1, 0, 0, 0, 0,...

%e . 22, 7, 6, 2, 2, 1, 1, 0, 1, 0, 0, 0,...

%e . 30, 15, 6, 5, 3, 2, 1, 1, 0, 1, 0, 0,...

%e . 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1, 0,...

%e . 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1,...

%e ...

%e For n = 7, from the conjecture we have that p(n-1) = p(6) = 11 = 3+8 = 2+3+6 = 1+3+2+5 = 1+1+2+3+4 = 0+1+1+2+2+5, etc. where p(n) = A000041(n).

%Y Columns 1-4: A000041, A182712, A182713, A182714. Main triangle: A182703.

%Y Cf. A066633, A135010, A138121, A138137.

%K nonn,tabl

%O 1,4

%A _Omar E. Pol_, Feb 04 2012