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%I #18 May 22 2012 12:03:03
%S 1,4,3,9,8,5,20,19,16,11,35,34,31,26,15,66,65,62,57,46,31,105,104,101,
%T 96,85,70,39,176,175,172,167,156,141,110,71,270,269,266,261,250,235,
%U 204,165,94,420,419,416,411,400,385,354,315,244,150,616,615
%N Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.
%C The set of partitions of n contains n shells (see A135010). It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.
%F T(n,k) = A066186(n) - A066186(k-1).
%F T(n,k) = Sum_{j=k..n} A138879(j).
%e For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below the sum of all parts of each set:
%e --------------------------------------------------------
%e . S{1-5} S{2-5} S{3-5} S{4-5} S{5}
%e --------------------------------------------------------
%e . The Last Last Last The
%e . five four three two last
%e . shells shells shells shells shell
%e . of 5 of 5 of 5 of 5 of 5
%e --------------------------------------------------------
%e .
%e . 5 5 5 5 5
%e . 3+2 3+2 3+2 3+2 3+2
%e . 4+1 4+1 4+1 4+1 1
%e . 2+2+1 2+2+1 2+2+1 2+2+1 1
%e . 3+1+1 3+1+1 3+1+1 1+1 1
%e . 2+1+1+1 2+1+1+1 1+1+1 1+1 1
%e . 1+1+1+1+1 1+1+1+1 1+1+1 1+1 1
%e . ---------- ---------- ---------- ---------- ----------
%e . 35 34 31 26 15
%e .
%e So row 5 lists 35, 34, 31, 26, 15.
%e .
%e Triangle begins:
%e 1;
%e 4, 3;
%e 9, 8, 5;
%e 20, 19, 16, 11;
%e 35, 34, 31, 26, 15;
%e 66, 65, 62, 57, 46, 31;
%e 105, 104, 101, 96, 85, 70, 39;
%e 176, 175, 172, 167, 156, 141, 110, 71;
%e 270, 269, 266, 261, 250, 235, 204, 165, 94;
%e 420, 419, 416, 411, 400, 385, 354, 315, 244, 150;
%Y Mirror of triangle A212011. Column 1 is A066186. Right border is A138879.
%Y Cf. A135010, A138121, A182703, A206563, A211980, A212000, A212010.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Apr 26 2012
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