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
Seiichi Manyama, Table of n, a(n) for n = 1..1000
FORMULA
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 13 2014
STATUS
approved