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A367944
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a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).
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4
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1, 6, 27216, 1344924798336, 3605580335899213007486976, 1648055031941075082958467426002632704000000, 312704667066499295437237787452750428210311485710262201221120000000
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OFFSET
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0,2
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COMMENTS
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In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).
c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.
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LINKS
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FORMULA
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a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...
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MATHEMATICA
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Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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