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A277235 Decimal expansion of 2/(Gamma(3/4))^4.

%I A277235 #26 Apr 26 2025 07:41:58
%S A277235 8,8,6,9,4,1,1,6,8,5,7,8,1,1,5,4,0,5,4,1,1,5,2,5,3,6,1,3,5,4,5,2,1,5,
%T A277235 3,8,6,8,6,4,9,9,9,1,9,6,4,2,5,9,8,3,4,8,3,0,9,8,6,0,9,8,9,8,1,3,1,7,
%U A277235 8,2,5,5,9,4,8,1,9,2,7,9,7,0,6,9,1,5,2,6,4,7,7,9,4,9,8,1,2,1
%N A277235 Decimal expansion of 2/(Gamma(3/4))^4.
%C A277235 This is the value of one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.
%C A277235 The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.
%D A277235 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.
%H A277235 G. C. Greubel, <a href="/A277235/b277235.txt">Table of n, a(n) for n = 0..10000</a>
%F A277235 Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
%F A277235 Equals A060294 * A091670.
%F A277235 For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).
%F A277235 From _Amiram Eldar_, Jul 13 2023: (Start)
%F A277235 Equals (Gamma(1/4)/Pi)^4/2.
%F A277235 Equals A088538 * A014549^2.
%F A277235 Equals A263809/Pi. (End)
%e A277235 2/Gamma(3/4)^4 = 0.88694116857811540541152...
%t A277235 RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* _G. C. Greubel_, Oct 26 2018 *)
%o A277235 (PARI) 2/gamma(3/4)^4 \\ _Michel Marcus_, Nov 13 2016
%o A277235 (Magma) SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // _G. C. Greubel_, Oct 26 2018
%Y A277235 Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.
%K A277235 nonn,cons
%O A277235 0,1
%A A277235 _Wolfdieter Lang_, Nov 13 2016