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%D S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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