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A255931
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a(n) is the numerator of Gamma(n+1/2)^2/(2*n*Pi), the value of an integral with sinh in the denominator.
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1
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1, 9, 75, 11025, 178605, 36018675, 2608781175, 4108830350625, 131939107925625, 85734032330071125, 17185776480709711875, 33334677780416604466875, 4807886218329317951953125, 6509191098729563747237109375
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OFFSET
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1,2
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LINKS
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FORMULA
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Integral_{-infinity..infinity} (prod_{j=1..n-1} j^2+x^2)*x/sinh(2*Pi*x) dx = Gamma(n+1/2)^2/(2*n*Pi).
The n-th fraction also equals the n-th coefficient in the expansion of 2F1(1/2,1/2; 1; x) * n!*(n-1)!/2.
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EXAMPLE
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1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...
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MATHEMATICA
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a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Numerator; Array[a, 15]
Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Numerator (* Vaclav Kotesovec, Mar 11 2015 *)
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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