A347869 -id:A347869 - cp's OEIS frontend

A347867 Number of partitions of n such that 3*(greatest part) >= (number of parts).

1, 2, 3, 4, 6, 10, 14, 20, 27, 38, 51, 70, 92, 123, 162, 212, 274, 355, 453, 579, 733, 928, 1165, 1463, 1822, 2269, 2808, 3470, 4266, 5241, 6407, 7823, 9514, 11554, 13983, 16900, 20359, 24494, 29386, 35205, 42069, 50206, 59773, 71069, 84322, 99913, 118157, 139556, 164528, 193734

Offset: 1

Seiichi Manyama, Jan 25 2022

nonn

Also, the number of partitions of n such that (greatest part) <= 3*(number of parts).

Cf. A064174, A237755, A347868, A347869.

Cf. A237756.

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(3*k+j-1))/(1-x^j))))

G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(3*k+j-1))/(1-x^j).

A347868 Number of partitions of n such that 4*(greatest part) >= (number of parts).

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 14, 21, 29, 40, 53, 73, 96, 129, 168, 221, 284, 369, 471, 603, 763, 966, 1211, 1521, 1892, 2355, 2912, 3600, 4423, 5434, 6639, 8107, 9855, 11968, 14476, 17495, 21067, 25342, 30393, 36406, 43489, 51891, 61761, 73421, 87087, 103172, 121977, 144045, 169780, 199883

Offset: 1

Seiichi Manyama, Jan 25 2022

nonn

Also, the number of partitions of n such that (greatest part) <= 4*(number of parts).

Cf. A064174, A237755, A347867, A347869.

Cf. A348163.

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(4*k+j-1))/(1-x^j))))

G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(4*k+j-1))/(1-x^j).

cp's OEIS Frontend

A347867 Number of partitions of n such that 3*(greatest part) >= (number of parts).

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