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A272854 Ramanujan's beta-series.

%I A272854 #21 Jul 04 2023 12:17:42
%S A272854 10,812,67402,5593538,464196268,38522696690,3196919629018,
%T A272854 265305806511788,22017185020849402,1827161050923988562,
%U A272854 151632350041670201260,12583657892407702716002
%N A272854 Ramanujan's beta-series.
%C A272854 Ramanujan's notes define this by the same G.f. as A051030 (the c-series) but using Laurent series expansion. It is mislabeled as "gamma" in Ramanujan's notes. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).
%D A272854 S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
%H A272854 Seiichi Manyama, <a href="/A272854/b272854.txt">Table of n, a(n) for n = 0..520</a>
%H A272854 Robert Munafo, <a href="http://www.mrob.com/pub/seq/ramanujan.html">Sequences Related to the Work of Srinivasa Ramanujan</a>
%H A272854 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (82, 82, -1).
%F A272854 G.f.: (10-8*x-2*x^2)/(1-82*x-82*x^2+x^3).
%F A272854 a(-3)=14258; a(-2)=172; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
%F A272854 A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.
%e A272854 a(3)=5593538 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
%t A272854 Rest@ CoefficientList[ Normal@Series[-(2 + 8*x - 10*x^2)/(1 - 82*x - 82*x^2 + x^3), {x, Infinity, 20}], 1/x] (* _Giovanni Resta_, May 08 2016 *)
%o A272854 (Wolfram|Alpha) Series[-1*(2+8a-10a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
%Y A272854 Cf. A051028, A051029, A051030.
%Y A272854 Cf. A272853, A272855.
%K A272854 nonn,easy
%O A272854 0,1
%A A272854 _Robert Munafo_, May 08 2016