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Sum of all region numbers of all parts of the last section of the set of partitions of n.
3

%I #17 Apr 17 2014 10:57:26

%S 1,4,9,29,55,157,277,669,1212,2555,4459,9048

%N Sum of all region numbers of all parts of the last section of the set of partitions of n.

%C Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

%C The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - _Omar E. Pol_, Apr 07 2014

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>

%e For n = 6 the four regions of the last section of 6 are [2], [4, 2], [3], [6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1] therefore the "region numbers" are [8], [9, 9], [10], [11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11]. The sum of all region numbers is a(6) = 8+2*9+10+11^2 = 8+18+10+121 = 157, see below:

%e --------------------------------------------

%e . Last section Sum of

%e . of the set of Region region

%e k partitions of 6 numbers numbers

%e --------------------------------------------

%e 11 6 11 11

%e 10 3+3 10,11 21

%e 9 4 +2 9, 11 20

%e 8 2+2 +2 8,9, 11 28

%e 7 1 11 11

%e 6 1 11 11

%e 5 1 11 11

%e 4 1 11 11

%e 3 1 11 11

%e 2 1 11 11

%e 1 1 11 11

%e --------------------------------------------

%e Total sum of region numbers is a(6) = 157

%Y Row sums of triangle A210966. Partial sums give A210972.

%Y Cf. A135010, A138121, A194446, A182703, A206437, A210971.

%K nonn,more

%O 1,2

%A _Omar E. Pol_, Jul 01 2012