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A272855 Ramanujan's gamma-series.

%I A272855 #23 Jul 04 2023 12:18:41
%S A272855 12,1010,83802,6954572,577145658,47896135058,3974802064140,
%T A272855 329860675188578,27374461238587818,2271750422127600332,
%U A272855 188527910575352239722,15645544827332108296610
%N A272855 Ramanujan's gamma-series.
%C A272855 Ramanujan's notes define this by the same G.f. as A051029 (the b-series) but using Laurent series expansion. It is mislabeled as "beta" 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%D A272855 S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
%H A272855 Seiichi Manyama, <a href="/A272855/b272855.txt">Table of n, a(n) for n = 0..520</a>
%H A272855 Robert Munafo, <a href="http://www.mrob.com/pub/seq/ramanujan.html">Sequences Related to the Work of Srinivasa Ramanujan</a>
%H A272855 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (82, 82, -1).
%F A272855 G.f.: x*(12+26*x-2*x^2)/(1-82*x-82*x^2+x^3).
%F A272855 a(-3)=11468; a(-2)=138; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
%F A272855 A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.
%e A272855 a(3)=6954572 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
%t A272855 Rest@ CoefficientList[ Normal@ Series[-1*(2 - 26 a - 12 a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 10}], 1/a] (* _Giovanni Resta_, May 08 2016 *)
%o A272855 (Wolfram|Alpha) Series[-1*(2-26a-12a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
%Y A272855 Cf. A051028, A051029, A051030.
%Y A272855 Cf. A272853, A272854.
%K A272855 nonn
%O A272855 0,1
%A A272855 _Robert Munafo_, May 08 2016