%I #22 Sep 08 2022 08:45:07

%S 1,32,32768,1048576,34359738368,1099511627776,1125899906842624,

%T 36028797018963968,37778931862957161709568,1208925819614629174706176,

%U 1237940039285380274899124224,39614081257132168796771975168

%N a(n) = denominator( (4*n+1)*(Product_{i=1..n} (2*i-1)/Product_{i=1..n} (2*i))^5 ).

%C For the partial sums of the series given in the formula section see A278140(n)/a(n). The value of the series is given in A277235. - _Wolfdieter Lang_, Nov 15 2016

%D Bruce C. Berndt and Robert Rankin,"Ramanujan: letters and commentary", AMS-LMS, History of Mathematics, vol. 9, p. 57

%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

%H Seiichi Manyama, <a href="/A074800/b074800.txt">Table of n, a(n) for n = 0..335</a>

%F a(n) = denominator(b(n)) with b(0) = 1 and b(n) = (4*n+1)*(Product_{i=1..n} (2*i-1) / Product_{i=1..n}(2*i))^5 = (4*n+1)*(A001147(n)/A000165(n))^5.

%F 1 + Sum_{k>=1} (-1)^k*b(k) = 2/Gamma(3/4)^4=0.88694116857811540541...(see

%F A277235).

%F a(n) = denominator( (4*n+1)*( binomial(2*n, n)/4^n )^5 ). - _G. C. Greubel_, Jul 09 2021

%t Table[Denominator[(4n+1) (Product[(2i-1), {i, n}]/Product[2i, {i, n}])^5], {n, 0, 10}] (* _Michael De Vlieger_, Nov 15 2016 *)

%o (PARI) a(n)=denominator ((4*n+1)*(prod(i=1,n,2*i-1)/prod(i=1,n,2*i))^5)

%o (Magma) [Denominator((4*n+1)*((n+1)*Catalan(n)/4^n)^5): n in [0..30]]; // _G. C. Greubel_, Jul 09 2021

%o (Sage) [denominator((4*n+1)*(binomial(2*n, n)/4^n)^5) for n in (0..30)] # _G. C. Greubel_, Jul 09 2021

%Y Cf. A074799, A277235, A278140.

%K easy,frac,nonn

%O 0,2

%A _Benoit Cloitre_, Sep 08 2002

%E Edited. - _Wolfdieter Lang_, Nov 15 2016