A284609 Number of partitions of n such that the (sum of all odd parts) = floor(n/2).
0, 0, 1, 1, 0, 0, 4, 4, 0, 0, 9, 12, 0, 0, 25, 30, 0, 0, 56, 70, 0, 0, 132, 165, 0, 0, 270, 330, 0, 0, 594, 704, 0, 0, 1140, 1380, 0, 0, 2268, 2688, 0, 0, 4256, 4984, 0, 0, 8008, 9394, 0, 0, 14342, 16665, 0, 0, 25920, 29970, 0, 0, 45056, 52096
Offset: 1
Examples
a(8) counts these 4 partitions: 431, 3221, 32111, 311111.
Programs
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Mathematica
Table[p = IntegerPartitions[n]; Length[Select[ Table[Total[Select[DeleteDuplicates[p[[k]]], EvenQ]], {k, Length[p]}], # == Floor[n/2] &]], {n, 60}] (* Peter J. C. Moses, Mar 29 2017 *)
Comments