cp's OEIS Frontend

A212001 Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.

%I A212001 #18 May 22 2012 12:03:03
%S A212001 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 A212001 96,85,70,39,176,175,172,167,156,141,110,71,270,269,266,261,250,235,
%U A212001 204,165,94,420,419,416,411,400,385,354,315,244,150,616,615
%N A212001 Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.
%C A212001 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 A212001 T(n,k) = A066186(n) - A066186(k-1).
%F A212001 T(n,k) = Sum_{j=k..n} A138879(j).
%e A212001 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 A212001 --------------------------------------------------------
%e A212001 .  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
%e A212001 --------------------------------------------------------
%e A212001 .  The        Last       Last       Last       The
%e A212001 .  five       four       three      two        last
%e A212001 .  shells     shells     shells     shells     shell
%e A212001 .  of 5       of 5       of 5       of 5       of 5
%e A212001 --------------------------------------------------------
%e A212001 .
%e A212001 .  5          5          5          5          5
%e A212001 .  3+2        3+2        3+2        3+2        3+2
%e A212001 .  4+1        4+1        4+1        4+1          1
%e A212001 .  2+2+1      2+2+1      2+2+1      2+2+1          1
%e A212001 .  3+1+1      3+1+1      3+1+1        1+1          1
%e A212001 .  2+1+1+1    2+1+1+1      1+1+1        1+1          1
%e A212001 .  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
%e A212001 . ---------- ---------- ---------- ---------- ----------
%e A212001 .     35         34         31         26         15
%e A212001 .
%e A212001 So row 5 lists 35, 34, 31, 26, 15.
%e A212001 .
%e A212001 Triangle begins:
%e A212001 1;
%e A212001 4,     3;
%e A212001 9,     8,   5;
%e A212001 20,   19,  16,  11;
%e A212001 35,   34,  31,  26,  15;
%e A212001 66,   65,  62,  57,  46,  31;
%e A212001 105, 104, 101,  96,  85,  70,  39;
%e A212001 176, 175, 172, 167, 156, 141, 110,  71;
%e A212001 270, 269, 266, 261, 250, 235, 204, 165,  94;
%e A212001 420, 419, 416, 411, 400, 385, 354, 315, 244, 150;
%Y A212001 Mirror of triangle A212011. Column 1 is A066186. Right border is A138879.
%Y A212001 Cf. A135010, A138121, A182703, A206563, A211980, A212000, A212010.
%K A212001 nonn,tabl
%O A212001 1,2
%A A212001 _Omar E. Pol_, Apr 26 2012