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A272854
Ramanujan's beta-series.
3
10, 812, 67402, 5593538, 464196268, 38522696690, 3196919629018, 265305806511788, 22017185020849402, 1827161050923988562, 151632350041670201260, 12583657892407702716002
OFFSET
0,1
COMMENTS
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).
REFERENCES
S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
FORMULA
G.f.: (10-8*x-2*x^2)/(1-82*x-82*x^2+x^3).
a(-3)=14258; a(-2)=172; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.
EXAMPLE
a(3)=5593538 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
MATHEMATICA
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 *)
PROG
(Wolfram|Alpha) Series[-1*(2+8a-10a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Munafo, May 08 2016
STATUS
approved