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A277234 Numerators of partial sums of a Ramanujan series converging to 2/Pi = A060294. 1
1, 3, 435, 1855, 1678635, 8178093, 831557727, 4362807735, 26663516457435, 146862472576105, 13439367283090749, 76661183599555737, 54390019021528255975, 318658997759516188425, 27581665786275463543575, 165068987339858265879975, 7173478080571052213369487675 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..16.

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.

EXAMPLE

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.

CROSSREFS

Cf. A060294, A241756, A277232.

Sequence in context: A172859 A087771 A332143 * A269553 A086207 A326373

Adjacent sequences:  A277231 A277232 A277233 * A277235 A277236 A277237

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 13 2016

STATUS

approved

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Last modified December 1 01:37 EST 2021. Contains 349426 sequences. (Running on oeis4.)