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A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x). 7
1, 9, 603, 4949, 2576763, 20864151, 1347632055, 10860010029, 44749069441659, 359788384157147, 23124997294306677, 185685617347012755, 95380005326947177879, 765237422887515344907, 49101291379356533433423, 393721549706169405868509, 12928613856208967961607217787 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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.

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.

EXAMPLE

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...

CROSSREFS

Cf. A001147, A000165, A091670, A241756.

Sequence in context: A332159 A139107 A226552 * A171703 A085530 A188423

Adjacent sequences:  A277229 A277230 A277231 * A277233 A277234 A277235

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 11 2016

STATUS

approved

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Last modified September 22 22:38 EDT 2021. Contains 347609 sequences. (Running on oeis4.)