cp's OEIS Frontend

A237754 Number of partitions of n such that 2*(greatest part) > (number of parts).

%I A237754 #20 May 22 2024 23:49:02
%S A237754 1,1,2,4,5,8,11,16,23,32,43,59,78,104,137,181,233,303,388,497,630,799,
%T A237754 1003,1262,1574,1961,2430,3008,3701,4551,5569,6805,8284,10070,12195,
%U A237754 14753,17786,21413,25709,30824,36856,44014,52435,62384,74062,87811,103901
%N A237754 Number of partitions of n such that 2*(greatest part) > (number of parts).
%C A237754 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.
%C A237754 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.
%H A237754 Seiichi Manyama, <a href="/A237754/b237754.txt">Table of n, a(n) for n = 1..1000</a>
%F A237754 a(n) = A000041(n) - A237752(n).
%F A237754 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
%e A237754 a(5) = 5 counts these partitions:  5, 41, 32, 311, 221.
%t A237754 z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] > Length[p]], {n, z}]
%o A237754 (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
%Y A237754 Cf. A064173, A237751-A237753, A237755-A237757, A000041.
%K A237754 nonn,easy
%O A237754 1,3
%A A237754 _Clark Kimberling_, Feb 13 2014