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%I A255931 #15 Mar 12 2015 05:18:38
%S A255931 1,9,75,11025,178605,36018675,2608781175,4108830350625,
%T A255931 131939107925625,85734032330071125,17185776480709711875,
%U A255931 33334677780416604466875,4807886218329317951953125,6509191098729563747237109375
%N A255931 a(n) is the numerator of Gamma(n+1/2)^2/(2*n*Pi), the value of an integral with sinh in the denominator.
%H A255931 MathOverflow, <a href="http://mathoverflow.net/questions/79093">An identity involving an infinite integral with a sinh in the denominator</a>
%F A255931 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).
%F A255931 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.
%e A255931 1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...
%t A255931 a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Numerator; Array[a, 15]
%t A255931 Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Numerator (* _Vaclav Kotesovec_, Mar 11 2015 *)
%Y A255931 Cf. A255932 (denominators).
%K A255931 frac,nonn
%O A255931 1,2
%A A255931 _Jean-François Alcover_, Mar 11 2015

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