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A237752
Number of partitions of n such that 2*(greatest part) <= (number of parts).
9
0, 1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 18, 23, 31, 39, 50, 64, 82, 102, 130, 162, 203, 252, 313, 384, 475, 580, 710, 864, 1053, 1273, 1544, 1859, 2240, 2688, 3224, 3851, 4602, 5476, 6514, 7727, 9160, 10826, 12791, 15072, 17747, 20853, 24481, 28679, 33577, 39231
OFFSET
1,5
COMMENTS
Also, the number of partitions of n such that (greatest part) >= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) <= 0.
Also, the number of partitions p of n such that max(max(p), 2*(number of parts of p)) is a part of p.
LINKS
FORMULA
a(n) = A000041(n) - A237754(n).
EXAMPLE
The partitions of 6 that do not qualify are 22311, 21111, 111111, so that a(6) = 11 - 3 = 8.
MATHEMATICA
z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] <= Length[p]], {n, z}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], 2*Length[p]]]], {n, 50}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 13 2014
STATUS
approved