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A212011 Triangle read by rows: T(n,k) = sum of all parts of the last k shells of n.

%I A212011 #15 May 22 2012 12:02:18
%S A212011 1,3,4,5,8,9,11,16,19,20,15,26,31,34,35,31,46,57,62,65,66,39,70,85,96,
%T A212011 101,104,105,71,110,141,156,167,172,175,176,94,165,204,235,250,261,
%U A212011 266,269,270,150,244,315,354,385,400,411,416,419,420,196,346
%N A212011 Triangle read by rows: T(n,k) = sum of all parts of the last k shells of n.
%C A212011 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 A212011 T(n,k) = A066186(n) - A066186(n-k).
%F A212011 T(n,k) = Sum_{j=n-k+1..n} A138879(j).
%e A212011 For n = 5 the illustration shows five sets containing the last k shells of 5 and below we can see that the sum of all parts of in each set:
%e A212011 --------------------------------------------------------
%e A212011 .  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
%e A212011 --------------------------------------------------------
%e A212011 .  The        Last       Last       Last       The
%e A212011 .  last       two        three      four       five
%e A212011 .  shell      shells     shells     shells     shells
%e A212011 .  of 5       of 5       of 5       of 5       of 5
%e A212011 --------------------------------------------------------
%e A212011 .
%e A212011 .  5          5          5          5          5
%e A212011 .  3+2        3+2        3+2        3+2        3+2
%e A212011 .    1        4+1        4+1        4+1        4+1
%e A212011 .      1      2+2+1      2+2+1      2+2+1      2+2+1
%e A212011 .      1        1+1      3+1+1      3+1+1      3+1+1
%e A212011 .        1        1+1      1+1+1    2+1+1+1    2+1+1+1
%e A212011 .          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
%e A212011 . ---------- ---------- ---------- ---------- ----------
%e A212011 .     15         26         31         34         35
%e A212011 .
%e A212011 So row 5 lists 15, 26, 31, 34, 35.
%e A212011 .
%e A212011 Triangle begins:
%e A212011 1;
%e A212011 3,     4;
%e A212011 5,     8,   9;
%e A212011 11,   16,  19,  20;
%e A212011 15,   26,  31,  34,  35;
%e A212011 31,   46,  57,  62,  65,  66;
%e A212011 39,   70,  85,  96, 101, 104, 105;
%e A212011 71,  110, 141, 156, 167, 172, 175, 176;
%e A212011 94,  165, 204, 235, 250, 261, 266, 269, 270;
%e A212011 150, 244, 315, 354, 385, 400, 411, 416, 419, 420;
%Y A212011 Mirror of triangle A212001. Column 1 is A138879. Right border is A066186.
%Y A212011 Cf. A135010, A182703, A212000, A212010.
%K A212011 nonn,tabl
%O A212011 1,2
%A A212011 _Omar E. Pol_, Apr 26 2012