A237755 - OEIS

%I #18 Jan 22 2022 20:59:49

%S 1,2,2,4,6,9,12,18,24,34,46,63,83,111,144,190,245,318,405,520,657,833,

%T 1045,1312,1634,2036,2517,3114,3829,4705,5751,7027,8544,10381,12564,

%U 15190,18301,22026,26425,31669,37849,45180,53796,63983,75923,89987,106435

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

%C 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 Seiichi Manyama, <a href="/A237755/b237755.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A000041(n) - A237751(n).

%F 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 a(6) = 9 counts all of the 11 partitions of 6 except these:  21111, 111111.

%t z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] >= Length[p]], {n, z}]

%o (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 for(n=1,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 03 2015

%Y Cf. A064173, A237751-A237755, A237756, A237757, A000041.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 13 2014