A370747 -id:A370747 - cp's OEIS frontend

A372703 Number of partitions of n into distinct parts such that number of parts is a multiples of 4.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 207, 226, 251, 273, 302, 328, 362, 393, 433, 470, 518, 563, 621, 677, 748, 818, 906, 994, 1104, 1216, 1354, 1497, 1671, 1853, 2073, 2305, 2582, 2877, 3226, 3599

Offset: 0

Seiichi Manyama, May 23 2024

nonn

Cf. A000009, A067661, A370747, A373078.

Cf. A363046.

my(N=70, x='x+O('x^N)); Vec(sum(k=0, N, prod(j=1, 4*k, x^j/(1-x^j))))

G.f.: Sum_{k>=0} x^(2*k*(4*k+1)) / Product_{j=1..4*k} (1-x^j) = Sum_{k>=0} Product_{j=1..4*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

A372703 Number of partitions of n into distinct parts such that number of parts is a multiples of 4.

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