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