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

1, 2, 2, 4, 6, 9, 12, 18, 24, 34, 46, 63, 83, 111, 144, 190, 245, 318, 405, 520, 657, 833, 1045, 1312, 1634, 2036, 2517, 3114, 3829, 4705, 5751, 7027, 8544, 10381, 12564, 15190, 18301, 22026, 26425, 31669, 37849, 45180, 53796, 63983, 75923, 89987, 106435

Offset: 1

Author

Clark Kimberling, Feb 13 2014

Keywords

• nonn
• easy

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.

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Mathematica

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

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)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015

Formula

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

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