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A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

%I A212000 #35 May 21 2012 15:57:56
%S A212000 1,3,2,6,5,3,12,11,9,6,20,19,17,14,8,35,34,32,29,23,15,54,53,51,48,42,
%T A212000 34,19,86,85,83,80,74,66,51,32,128,127,125,122,116,108,93,74,42,192,
%U A212000 191,189,186,180,172,157,138,106,64,275,274,272,269,263,255,240
%N A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.
%C A212000 The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.
%C A212000 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 A212000 T(n,k) = A006128(n) - A006128(k-1).
%F A212000 T(n,k) = Sum_{j=k..n} A138137(j).
%e A212000 For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
%e A212000 --------------------------------------------------------
%e A212000 .  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
%e A212000 --------------------------------------------------------
%e A212000 .  The        Last       Last       Last       The
%e A212000 .  five       four       three      two        last
%e A212000 .  shells     shells     shells     shells     shell
%e A212000 .  of 5       of 5       of 5       of 5       of 5
%e A212000 --------------------------------------------------------
%e A212000 .
%e A212000 .  5          5          5          5          5
%e A212000 .  3+2        3+2        3+2        3+2        3+2
%e A212000 .  4+1        4+1        4+1        4+1          1
%e A212000 .  2+2+1      2+2+1      2+2+1      2+2+1          1
%e A212000 .  3+1+1      3+1+1      3+1+1        1+1          1
%e A212000 .  2+1+1+1    2+1+1+1      1+1+1        1+1          1
%e A212000 .  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
%e A212000 . ---------- ---------- ---------- ---------- ----------
%e A212000 . 20         19         17         14          8
%e A212000 .
%e A212000 So row 5 lists 20, 19, 17, 14, 8.
%e A212000 .
%e A212000 Triangle begins:
%e A212000 1;
%e A212000 3,     2;
%e A212000 6,     5,   3;
%e A212000 12,   11,   9,   6;
%e A212000 20,   19,  17,  14,  8;
%e A212000 35,   34,  32,  29,  23,  15;
%e A212000 54,   53,  51,  48,  42,  34,  19;
%e A212000 86,   85,  83,  80,  74,  66,  51,  32;
%e A212000 128, 127, 125, 122, 116, 108,  93,  74,  42;
%e A212000 192, 191, 189, 186, 180, 172, 157, 138, 106, 64;
%Y A212000 Mirror of triangle A212010. Column 1 is A006128. Right border gives A138137.
%Y A212000 Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011
%K A212000 nonn,tabl
%O A212000 1,2
%A A212000 _Omar E. Pol_, Apr 26 2012