A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x).

1, 9, 603, 4949, 2576763, 20864151, 1347632055, 10860010029, 44749069441659, 359788384157147, 23124997294306677, 185685617347012755, 95380005326947177879, 765237422887515344907, 49101291379356533433423, 393721549706169405868509, 12928613856208967961607217787

Offset: 0

Author

Wolfdieter Lang, Nov 11 2016

Keywords

• nonn
• frac
• easy

Comments

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.

Examples

			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...
		

References

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

Links

• 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.

Crossrefs

Cf. A001147, A000165, A091670, A241756.

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.