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%I #33 Oct 02 2013 15:58:41
%S 1,2,1,3,1,1,5,1,1,1,7,1,1,1,1,11,1,1,1,1,1,15,1,1,1,1,1,1,22,1,1,1,1,
%T 1,1,1,30,1,1,1,1,1,1,1,1,42,1,1,1,1,1,1,1,1,1,56,1,1,1,1,1,1,1,1,1,1,
%U 77,1,1,1,1,1,1,1,1,1,1,1,101,1,1,1
%N Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.
%C The sum of row n is S_n = n - 1 + A000041(n) = A133041(n) - 1.
%C Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k) is also the number of k-th parts of all partitions of n in the k-th column of rectangle.
%e Triangle begins:
%e 1;
%e 2, 1;
%e 3, 1, 1;
%e 5, 1, 1, 1;
%e 7, 1, 1, 1, 1;
%e 11, 1, 1, 1, 1, 1;
%e 15, 1, 1, 1, 1, 1, 1;
%e 22, 1, 1, 1, 1, 1, 1, 1;
%e 30, 1, 1, 1, 1, 1, 1, 1, 1;
%e 42, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%Y Main diagonal of A209655 and of A209918.
%Y Cf. A000041, A008284, A058399, A133041, A135010, A141285, A209656, A207380.
%K nonn,tabl,easy
%O 1,2
%A _Omar E. Pol_, Mar 26 2012
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