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%I #21 Sep 08 2022 08:45:07
%S 1,5,2187,40625,892871875,20841167403,16443713753775,421390226721321,
%T 364130196991193221875,9816949116755633084375,
%U 8619392462988365485907909,239904481399203205153660455
%N a(n) = numerator( (4*n+1)*(Product_{i=1..n} (2*i-1)/Product_{i=1..n} 2*i)^5 ).
%D Bruce C. Berndt and Robert Rankin, "Ramanujan : letters and commentary", AMS-LMS, History of mathematics vol. 9, p. 57
%H Seiichi Manyama, <a href="/A074799/b074799.txt">Table of n, a(n) for n = 0..335</a>
%F a(n) = numerator of (b(n)), where b(n) = (4*n+1)*(Product_{i=1..n} (2*i - 1)/Product_{i=1..n} 2*i)^5 and b(0) = 1.
%F 1 + Sum_{k>=1} (-1)^k*b(k) = 2/gamma(3/4)^4 = 0.88694116857811540541...
%F a(n) = numerator( (4*n+1)*( binomial(2*n, n)/4^n )^5 ). - _G. C. Greubel_, Jul 09 2021
%t Table[Numerator[(4*n+1)*(Binomial[2*n, n]/4^n)^5], {n,0,30}] (* _G. C. Greubel_, Jul 09 2021 *)
%o (PARI) a(n)=numerator((4*n+1)*(prod(i=1,n,2*i-1)/prod(i=1,n,2*i))^5)
%o (Magma) [Numerator((4*n+1)*((n+1)*Catalan(n)/4^n)^5): n in [0..30]]; // _G. C. Greubel_, Jul 09 2021
%o (Sage) [numerator((4*n+1)*(binomial(2*n, n)/4^n)^5) for n in (0..30)] # _G. C. Greubel_, Jul 09 2021
%Y Cf. A074800 (denominators).
%K easy,frac,nonn
%O 0,2
%A _Benoit Cloitre_, Sep 08 2002
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