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A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

%I A212010 #58 May 21 2012 15:57:31
%S A212010 1,2,3,3,5,6,6,9,11,12,8,14,17,19,20,15,23,29,32,34,35,19,34,42,48,51,
%T A212010 53,54,32,51,66,74,80,83,85,86,42,74,93,108,116,122,125,127,128,64,
%U A212010 106,138,157,172,180,186,189,191,192,83,147,189,221,240
%N A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.
%C A212010 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 A212010 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 A212010 T(n,k) = A006128(n) - A006128(n-k).
%F A212010 T(n,k) = Sum_{j=n-k+1..n} A138137(j).
%e A212010 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 the first column equals the total number of parts in each set:
%e A212010 --------------------------------------------------------
%e A212010 .  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
%e A212010 --------------------------------------------------------
%e A212010 .  The        Last       Last       Last       The
%e A212010 .  last       two        three      four       five
%e A212010 .  shell      shells     shells     shells     shells
%e A212010 .  of 5       of 5       of 5       of 5       of 5
%e A212010 --------------------------------------------------------
%e A212010 .
%e A212010 .  5          5          5          5          5
%e A212010 .  3+2        3+2        3+2        3+2        3+2
%e A212010 .    1        4+1        4+1        4+1        4+1
%e A212010 .      1      2+2+1      2+2+1      2+2+1      2+2+1
%e A212010 .      1        1+1      3+1+1      3+1+1      3+1+1
%e A212010 .        1        1+1      1+1+1    2+1+1+1    2+1+1+1
%e A212010 .          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
%e A212010 . ---------- ---------- ---------- ---------- ----------
%e A212010 .  8         14         17         19         20
%e A212010 .
%e A212010 So row 5 lists 8, 14, 17, 19, 20.
%e A212010 .
%e A212010 Triangle begins:
%e A212010 1;
%e A212010 2,    3;
%e A212010 3,    5,   6;
%e A212010 6,    9,  11,  12;
%e A212010 8,   14,  17,  19,  20;
%e A212010 15,  23,  29,  32,  34,  35;
%e A212010 19,  34,  42,  48,  51,  53,  54;
%e A212010 32,  51,  66,  74,  80,  83,  85,  86;
%e A212010 42,  74,  93, 108, 116, 122, 125, 127, 128;
%e A212010 64, 106, 138, 157, 172, 180, 186, 189, 191, 192;
%Y A212010 Mirror of triangle A212000. Column 1 is A138137. Right border is A006128.
%Y A212010 Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011
%K A212010 nonn,tabl
%O A212010 1,2
%A A212010 _Omar E. Pol_, Apr 26 2012