A291721 -id:A291721 - cp's OEIS frontend

A291720 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^(k^3).

1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9, 9, 9, 45, 45, 45, 45, 45, 45, 45, 45, 165, 165, 165, 192, 192, 192, 192, 192, 522, 522, 522, 738, 738, 738, 738, 738, 1530, 1530, 1530, 2502, 2502, 2502, 2502, 2502, 4218, 4218, 4218, 7458, 7458, 7458, 7836, 7836

Offset: 0

Vaclav Kotesovec, Aug 30 2017

nonn

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

Cf. A000219, A003108, A291655, A291692, A291721.

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

log(a(n)) ~ 7 * (Gamma(1/3) * Zeta(7/3))^(3/7) * n^(4/7) / (2^(8/7) * 3^(9/7)).

A291666 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^(k^2).

Original entry on oeis.org

1, 2, 2, 2, 10, 18, 18, 18, 50, 100, 118, 118, 206, 438, 582, 582, 806, 1606, 2344, 2506, 3122, 5322, 8202, 9498, 11130, 16844, 26110, 32272, 37018, 52274, 78018, 100098, 115986, 155026, 223190, 291674, 345132, 439518, 618734, 811790, 972846, 1204190, 1653726

Offset: 0

Vaclav Kotesovec, Aug 29 2017

nonn

Convolution of A291649 and A291655.

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

Cf. A001156, A033461, A103265, A156616, A206622, A291649, A291655, A291667, A291721.

nmax = 50; CoefficientList[Series[Product[((1+x^(k^2))/(1-x^(k^2)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ 5 * Pi^(1/5) * ((8-sqrt(2)) * Zeta(5/2))^(2/5) * n^(3/5) / (4*3^(3/5)).

cp's OEIS Frontend

A291720 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^(k^3).

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