A372703 -id:A372703 - cp's OEIS frontend

A370747 Number of partitions of n into distinct parts such that number of parts is a multiples of 3.

1, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 28, 31, 35, 40, 45, 51, 59, 66, 76, 87, 100, 114, 133, 151, 175, 201, 232, 265, 307, 349, 402, 458, 524, 594, 680, 767, 872, 983, 1112, 1248, 1409, 1575, 1770, 1976, 2211, 2460, 2748, 3048, 3393, 3759, 4173, 4612, 5112

Offset: 0

Seiichi Manyama, May 23 2024

nonn

			a(12) = 7 counts these partitions: 921, 831, 741, 732, 651, 642, 543.

Cf. A000009, A067661, A372703, A373078.

Cf. A363045.

my(N=70, x='x+O('x^N)); Vec(sum(k=0, N, prod(j=1, 3*k, x^j/(1-x^j))))

G.f.: Sum_{k>=0} x^(3*k*(3*k+1)/2) / Product_{j=1..3*k} (1-x^j) = Sum_{k>=0} Product_{j=1..3*k} (x^j/(1-x^j)).

A373078 Number of partitions of n into distinct parts such that number of parts is a multiples of 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 141, 164, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1748, 1899, 2064, 2236, 2423, 2618, 2829, 3049, 3288, 3537, 3807

Offset: 0

Seiichi Manyama, May 23 2024

nonn

Cf. A000009, A067661, A370747, A372703.

Cf. A363047.

my(N=70, x='x+O('x^N)); Vec(sum(k=0, N, prod(j=1, 5*k, x^j/(1-x^j))))

G.f.: Sum_{k>=0} x^(5*k*(5*k+1)/2) / Product_{j=1..5*k} (1-x^j) = Sum_{k>=0} Product_{j=1..5*k} (x^j/(1-x^j)).

cp's OEIS Frontend

A370747 Number of partitions of n into distinct parts such that number of parts is a multiples of 3.

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