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

%I A277232 #30 Feb 16 2025 08:33:36
%S A277232 1,9,603,4949,2576763,20864151,1347632055,10860010029,44749069441659,
%T A277232 359788384157147,23124997294306677,185685617347012755,
%U A277232 95380005326947177879,765237422887515344907,49101291379356533433423,393721549706169405868509,12928613856208967961607217787
%N A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x).
%C A277232 The denominators seem to coincide with A241756.
%C A277232 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.
%C A277232 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.
%D A277232 G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 104.
%H A277232 Seiichi Manyama, <a href="/A277232/b277232.txt">Table of n, a(n) for n = 0..555</a>
%H A277232 F. Morley, <a href="https://doi.org/10.1112/plms/s1-34.1.397">On the Series 1 + (p/1)^3 + {p*(p+1)/1.2}^3 + ... </a>, Proc. London Math. Soc. 34 (1902) 397-402, eq. (5), p. 401.
%H A277232 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MorleysFormula.html">Morley's Formula</a>.
%F A277232 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.
%e A277232 The rationals r(n) begin: 1, 9/8, 603/512, 4949/4096, 2576763/2097152, 20864151/16777216, 1347632055/1073741824, ...
%e A277232 The limit is given in A091670, approximately 1.3932039296856768591...
%Y A277232 Cf. A001147, A000165, A091670, A241756.
%K A277232 nonn,frac,easy
%O A277232 0,2
%A A277232 _Wolfdieter Lang_, Nov 11 2016