A291720 -id:A291720 - cp's OEIS frontend

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

1, 1, 1, 1, 5, 5, 5, 5, 15, 24, 24, 24, 44, 80, 80, 80, 131, 221, 266, 266, 386, 566, 746, 746, 990, 1474, 1924, 2089, 2529, 3709, 4609, 5269, 6130, 8576, 11096, 12746, 14937, 19397, 25697, 28997, 34111, 43135, 56365, 65905, 76219, 95770, 120070, 144370, 163661

Offset: 0

Vaclav Kotesovec, Aug 28 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001156, A033461, A281790, A285047, A291649, A291696, A291720.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(k^2))^(k^2), {k,1,nmax}], {x,0,nmax}], x]

log(a(n)) ~ 5 * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5) / (2^(6/5) * 3^(3/5)).

A291721 Expansion of Product_{k>=1} ((1 + x^(k^3))/(1 - x^(k^3)))^(k^3).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 18, 34, 34, 34, 34, 34, 34, 34, 162, 290, 290, 290, 290, 290, 290, 290, 978, 1666, 1666, 1720, 1774, 1774, 1774, 1774, 4590, 7406, 7406, 8270, 9134, 9134, 9134, 9134, 18558, 27982, 27982, 34894, 41806, 41806, 41806, 41806, 68814, 95822

Offset: 0

Vaclav Kotesovec, Aug 30 2017

nonn

Convolution of A291692 and A291720.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A156616, A291666, A291692, A291720.

nmax = 100; CoefficientList[Series[Product[((1 + x^(k^3))/(1 - x^(k^3)))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ 7 * ((2^(7/3)-1) * Gamma(1/3) * Zeta(7/3))^(3/7) * n^(4/7) / (2^(12/7) * 3^(9/7)).

A303168 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 10, 10, 10, 13, 13, 13, 13, 13, 18, 18, 18, 24, 24, 24, 24, 24, 30, 30, 30, 39, 39, 39, 39, 39, 46, 46, 46, 58, 58, 58, 64, 64, 72, 72, 72, 87, 87, 87, 99, 99, 112, 112, 112, 130, 130, 130, 148, 148, 166, 166, 166, 187

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

Number of partitions of n into 1 kind of part 1, 2 kinds of part 8, 3 kinds of part 27, ..., k kinds of part k^3.

Cf. A000219, A000578, A003108, A023872, A285047, A291720, A298434, A300975.

nmax = 75; CoefficientList[Series[Product[1/(1 - x^k^3)^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^A000578(k))^k.

cp's OEIS Frontend

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

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A291721 Expansion of Product_{k>=1} ((1 + x^(k^3))/(1 - x^(k^3)))^(k^3).

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A303168 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^k.

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