cp's OEIS Frontend

A237755 Number of partitions of n such that 2*(greatest part) >= (number of parts).

%I A237755 #18 Jan 22 2022 20:59:49
%S A237755 1,2,2,4,6,9,12,18,24,34,46,63,83,111,144,190,245,318,405,520,657,833,
%T A237755 1045,1312,1634,2036,2517,3114,3829,4705,5751,7027,8544,10381,12564,
%U A237755 15190,18301,22026,26425,31669,37849,45180,53796,63983,75923,89987,106435
%N A237755 Number of partitions of n such that 2*(greatest part) >= (number of parts).
%C A237755 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.
%H A237755 Seiichi Manyama, <a href="/A237755/b237755.txt">Table of n, a(n) for n = 1..1000</a>
%F A237755 a(n) = A000041(n) - A237751(n).
%F A237755 G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(2*n+k-1))/(1 - x^k). - _Paul D. Hanna_, Aug 03 2015
%e A237755 a(6) = 9 counts all of the 11 partitions of 6 except these:  21111, 111111.
%t A237755 z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] >= Length[p]], {n, z}]
%o A237755 (PARI) {a(n) = my(A); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(2*m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
%o A237755 for(n=1,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 03 2015
%Y A237755 Cf. A064173, A237751-A237755, A237756, A237757, A000041.
%K A237755 nonn,easy
%O A237755 1,2
%A A237755 _Clark Kimberling_, Feb 13 2014