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a(n) = Product_{k=0..n} binomial(n^2, k^2).
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4
1, 1, 4, 1134, 333132800, 1319947441510156250, 876533819183888230348458418944000, 1185269534290897564185384010731432113450477770983533184
FORMULA
a(n) ~ c * exp(2*n*(2*n^2/3 + 1)) / (A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(7*n/6 - 1/4)), where c = 0.6367427... and A is the Glaisher-Kinkelin constant A074962.
MATHEMATICA
Table[Product[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 8}]
a(n) = Product_{k=0..n} (n^2 + k^2)!.
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4
1, 2, 116121600, 52498561358549216844165257625600000000
COMMENTS
The next term has 107 digits.
FORMULA
a(n) ~ 2^(4*n^3/3 + n^2 + 7*n/6 + 3/4) * exp(-26*n^3/9 + Pi*n^3/3 - 3*n^2/2 + Pi*n/4 - n) * Pi^((n+1)/2) * n^(8*n^3/3 + 3*n^2 + 4*n/3 + 1).
MATHEMATICA
Table[Product[(n^2+k^2)!, {k, 0, n}], {n, 0, 5}]
a(n) = Product_{k=0..n} (n^2 + k^2)! / (n^2 - k^2)!.
+10
2
1, 2, 806400, 29900785676206001356800000, 1118776785681133797769642926006209350326602179759885516800000000000000
FORMULA
a(n) ~ c * 2^(n^2 - n/6 + 1/4) * exp((3*Pi-10)*n^3/9 - n^2 + Pi*n/4) * n^(4*n^3/3 + 2*n^2 + n/2 + 3/4) / A^(2*n), where c = 1.941002... = A255504 * (c from A371603) and A is the Glaisher-Kinkelin constant A074962.
MATHEMATICA
Table[Product[(n^2+k^2)!/(n^2-k^2)!, {k, 0, n}], {n, 0, 6}]
a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2 - k^2).
+10
0
1, 1, 10, 57915, 8235313944000, 1077099640691257742845893750, 4629575796245443900868634734946423885068807034000
FORMULA
a(n) ~ c * exp(Pi*n^3/3 + Pi*n/4 + n) / (2^(2*n^3/3 + 3*n/2) * Pi^(n/2) * A^(2*n) * n^(7*n/6 - 1/4)), where c = 0.761512... = 2^(1/4) * A255504 * (c from A371603) / (c from A371645) and A is the Glaisher-Kinkelin constant A074962.
MATHEMATICA
Table[Product[Binomial[n^2+k^2, n^2-k^2], {k, 0, n}], {n, 0, 8}]
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