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A272855 Ramanujan's gamma-series. 3
12, 1010, 83802, 6954572, 577145658, 47896135058, 3974802064140, 329860675188578, 27374461238587818, 2271750422127600332, 188527910575352239722, 15645544827332108296610 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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).
REFERENCES
S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
LINKS
FORMULA
G.f.: x*(12+26*x-2*x^2)/(1-82*x-82*x^2+x^3).
a(-3)=11468; a(-2)=138; 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)=6954572 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
MATHEMATICA
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 *)
PROG
(Wolfram|Alpha) Series[-1*(2-26a-12a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
CROSSREFS
Sequence in context: A204327 A370313 A123283 * A365571 A265488 A232912
KEYWORD
nonn
AUTHOR
Robert Munafo, May 08 2016
STATUS
approved

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Last modified July 25 07:50 EDT 2024. Contains 374586 sequences. (Running on oeis4.)