A272855 Ramanujan's gamma-series.

12, 1010, 83802, 6954572, 577145658, 47896135058, 3974802064140, 329860675188578, 27374461238587818, 2271750422127600332, 188527910575352239722, 15645544827332108296610

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S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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

Seiichi Manyama, Table of n, a(n) for n = 0..520

Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan

Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

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

Cf. A051028, A051029, A051030.

Cf. A272853, A272854.

Sequence in context: A204327 A370313 A123283 * A365571 A265488 A232912

Adjacent sequences: A272852 A272853 A272854 * A272856 A272857 A272858

Keyword

nonn

Author

Robert Munafo, May 08 2016

Status

approved