A277234 Numerators of partial sums of a Ramanujan series converging to 2/Pi = A060294.

1, 3, 435, 1855, 1678635, 8178093, 831557727, 4362807735, 26663516457435, 146862472576105, 13439367283090749, 76661183599555737, 54390019021528255975, 318658997759516188425, 27581665786275463543575, 165068987339858265879975, 7173478080571052213369487675

Offset: 0

Author

Wolfdieter Lang, Nov 13 2016

Keywords

• nonn
• frac
• easy

Comments

The denominators seem to be A241756.

One of Ramanujan's series is 1 - 5*(1/2)^3 + 9*(1*3/(2*4))^3 - 13*(1*3*5/(2*4*6))^3 +- ... = Sum_{k>=0} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^3 where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.2) and p. 105, eq. (7.4.2) for s=1/2. The limit is Buffon's constant 2/Pi given in A060294.

Examples

			The rationals r(n) begin: 1, 3/8, 435/512, 1855/4096, 1678635/2097152, 8178093/16777216, 831557727/1073741824, 4362807735/8589934592, ...
The limit r(n), n -> oo, is 2/Pi = 0.6366197723... given in A060294.
		

References

• G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105.

Crossrefs

Cf. A060294, A241756, A277232.

Formula

a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^3 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^3. The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.