A304459 -id:A304459 - cp's OEIS frontend

A304460 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^3).

1, 1, 44, 4410, 905840, 318906400, 172185088824, 132357574570221, 137406570363495360, 185242628827767255255, 314645673306845990409300, 657405676947400829561901359, 1656968286301847988118098735168, 4957425610652588047512198547937050

Offset: 0

Vaclav Kotesovec, May 13 2018

nonn

In general, for m>=3, coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^m) is asymptotic to n^(m*n)/n!.

Cf. A008485, A023872, A304446, A304459.

nmax = 20; Table[SeriesCoefficient[Product[1/(1-x^k)^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[1/QPochhammer[x]^(n^3), {x, 0, n}], {n, 0, nmax}]

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).

A304461 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^3).

Original entry on oeis.org

1, 2, 144, 29232, 12263552, 8807437800, 9671073636672, 15075101792958592, 31660212257148109824, 86182291753025176234602, 295133367252867736074882400, 1241742977667269060006125955952, 6296492342467004634980003629748736, 37869525230334631809014462278624137096

Offset: 0

Vaclav Kotesovec, May 13 2018

nonn

In general, for m>=3, coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^m) is asymptotic to 2^n * n^(m*n) / n!.

Cf. A206623, A270919, A304448, A304459, A304460.

nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(n^3), {x, 0, n}], {n, 0, nmax}]

a(n) ~ 2^(n - 1/2) * exp(n) * n^(2*n - 1/2) / sqrt(Pi).

cp's OEIS Frontend

A304460 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^3).

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