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

1,2

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.

Links

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A340496 A035564 A240065 * A362260 A306730 A209603

Adjacent sequences: A237752 A237753 A237754 * A237756 A237757 A237758

Keyword

nonn,easy

Author

Clark Kimberling, Feb 13 2014

Status

approved