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

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, 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, 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

Offset: 0

Author

Wolfdieter Lang, Nov 13 2016

Keywords

• nonn
• cons

Comments

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.

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.

Examples

			2/Gamma(3/4)^4 = 0.88694116857811540541152...
		

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.

Programs

Magma

SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018

Mathematica

RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

PARI

2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

Formula

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.

Equals A060294 * A091670.

For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).

From Amiram Eldar, Jul 13 2023: (Start)

Equals (Gamma(1/4)/Pi)^4/2.

Equals A088538 * A014549^2.

Equals A263809/Pi. (End)