0,2

The denominators seem to coincide with A241756.

These are the partial sums of F. Morley's series Sum_{k>=0} (risefac(m,k)/k!)^3 for m=1/2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, pp. 104, 111.

The Morley formula gives the value of this series for |m| < 2/3 as Gamma(1-3*m/2)/(Gamma(1-m/2)^3)*cos(Pi*m/2). For the present case m=1/2 this value is hypergeometric([1/2,1/2,1/2],[1,1],1) = Pi/Gamma(3/4)^4 given in A091670.

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 104.

Seiichi Manyama, Table of n, a(n) for n = 0..555

F. Morley, On the Series 1 + (p/1)^3 + {p*(p+1)/1.2}^3 + ... , Proc. London Math. Soc. 34 (1902) 397-402, eq. (5), p. 401.

Eric Weisstein's World of Mathematics, Morley's Formula.

a(n) = numerator(r(n)) with the rational r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (-1)^k*(binomial(-1/2,k))^3 = Sum_{k=0..n} ((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.

The rationals r(n) begin: 1, 9/8, 603/512, 4949/4096, 2576763/2097152, 20864151/16777216, 1347632055/1073741824, ...

The limit is given in A091670, approximately 1.3932039296856768591...

Cf. A001147, A000165, A091670, A241756.

Sequence in context: A332159 A139107 A226552 * A171703 A352276 A085530

Adjacent sequences: A277229 A277230 A277231 * A277233 A277234 A277235

nonn,frac,easy

Wolfdieter Lang, Nov 11 2016

approved