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A237754
Number of partitions of n such that 2*(greatest part) > (number of parts).
6
1, 1, 2, 4, 5, 8, 11, 16, 23, 32, 43, 59, 78, 104, 137, 181, 233, 303, 388, 497, 630, 799, 1003, 1262, 1574, 1961, 2430, 3008, 3701, 4551, 5569, 6805, 8284, 10070, 12195, 14753, 17786, 21413, 25709, 30824, 36856, 44014, 52435, 62384, 74062, 87811, 103901
OFFSET
1,3
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 not a part of p.
LINKS
FORMULA
a(n) = A000041(n) - A237752(n).
G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(2*k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022
EXAMPLE
a(5) = 5 counts these partitions: 5, 41, 32, 311, 221.
MATHEMATICA
z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] > Length[p]], {n, z}]
PROG
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(2*k+j-2))/(1-x^j)))) \\ Seiichi Manyama, Jan 25 2022
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 13 2014
STATUS
approved