A237752 - OEIS

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A237752

Number of partitions of n such that 2*(greatest part) <= (number of parts).

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

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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

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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