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A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.

%I A208482 #54 Mar 30 2012 17:37:59
%S A208482 0,1,1,2,1,1,4,1,2,1,7,1,3,2,1,12,2,5,4,2,1,18,3,6,6,4,2,1,29,6,9,10,
%T A208482 7,4,2,1,42,9,11,13,11,7,4,2,1,63,16,15,19,17,12,7,4,2,1,89,24,18,25,
%U A208482 24,18,12,7,4,2,1,128,39,24,36,34,28,19,12,7,4,2,1
%N A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.
%C A208482 For the definition of the k-th rank see A208478.
%C A208482 It appears that the sum of the k-th ranks of all partitions of n is equal to zero.
%C A208482 It appears that reversed rows converge to A000070, the same as A208478. - Omar E. Pol, Mar 10 2012
%H A208482 Alois P. Heinz, <a href="/A208482/b208482.txt">Rows n = 1..44, flattened</a>
%e A208482 For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
%e A208482 ----------------------------------------------------------
%e A208482 Partitions    First      Second       Third      Fourth
%e A208482 of 4          rank        rank        rank        rank
%e A208482 ----------------------------------------------------------
%e A208482 4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
%e A208482 3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
%e A208482 2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
%e A208482 2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
%e A208482 1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
%e A208482 ----------------------------------------------------------
%e A208482 The sums of positive k-th ranks of the partitions of 4 are 4, 1, 2, 1 so row 4 lists 4, 1, 2, 1.
%e A208482 Triangle begins:
%e A208482 0;
%e A208482 1,    1;
%e A208482 2,    1,  1;
%e A208482 4,    1,  2,  1;
%e A208482 7,    1,  3,  2,  1;
%e A208482 12,   2,  5,  4,  2,  1;
%e A208482 18,   3,  6,  6,  4,  2,  1;
%e A208482 29,   6,  9, 10,  7,  4,  2,  1;
%e A208482 42,   9, 11, 13, 11,  7,  4,  2,  1;
%e A208482 63,  16, 15, 19, 17, 12,  7,  4,  2,  1;
%e A208482 89,  24, 18, 25, 24, 18, 12,  7,  4,  2,  1;
%e A208482 128, 39, 24, 36, 34, 28, 19, 12,  7,  4,  2,  1;
%Y A208482 Column 1 is A209616. Row sums give A208483.
%Y A208482 Cf. A006128, A063995, A064173, A105805, A181187, A194547, A194549, A195822, A208478.
%K A208482 nonn,tabl
%O A208482 1,4
%A A208482 _Omar E. Pol_, Mar 07 2012
%E A208482 Terms a(1)-a(22) confirmed and additional terms added by _John W. Layman_, Mar 10 2012