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#22 by Vaclav Kotesovec at Fri Dec 08 04:44:51 EST 2023
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#21 by Vaclav Kotesovec at Fri Dec 08 04:44:44 EST 2023
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Cf. A079478, (m=1), A324403, (m=2), A324426, (m=3), A324437, (m=4), A324438, (m=5), A324439, (m=6), A324440. (m=7).
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approved
editing
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#20 by Vaclav Kotesovec at Thu Dec 07 10:48:08 EST 2023
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#19 by Vaclav Kotesovec at Thu Dec 07 10:46:33 EST 2023
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| COMMENTS
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For m>1 > 0, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4 - v)), where v = 0 if m > 1 and v = 1/6 if m = 1, s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m).
c(1) = A / (2^(1/12) * exp(1/12) * sqrt(Pi)).
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approved
editing
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#18 by Vaclav Kotesovec at Thu Dec 07 10:35:42 EST 2023
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#17 by Vaclav Kotesovec at Thu Dec 07 10:35:36 EST 2023
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#16 by Vaclav Kotesovec at Thu Dec 07 10:34:39 EST 2023
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#15 by Vaclav Kotesovec at Thu Dec 07 10:34:34 EST 2023
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| COMMENTS
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c(4) = A306620.
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approved
editing
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#14 by Vaclav Kotesovec at Thu Dec 07 10:31:29 EST 2023
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#13 by Vaclav Kotesovec at Thu Dec 07 10:31:10 EST 2023
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c(2) = exp(Pi/12) * Gamma(1/4) / (2^(5/4) * Pi^(5/4)).
c(3) = A * 3^(1/6) * exp(Pi/(6*sqrt(3)) - 1/12) * Gamma(1/3)^2 / (2^(7/4) * Pi^(13/6)), where A = A074962 is the Glaisher-Kinkelin constant.
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approved
editing
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