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A261454
Expansion of a(x^2) / f(-x) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function.
1
1, 1, 8, 9, 17, 25, 47, 63, 106, 144, 216, 296, 425, 569, 807, 1064, 1449, 1905, 2551, 3304, 4353, 5592, 7254, 9247, 11859, 14978, 19038, 23872, 30034, 37433, 46734, 57854, 71739, 88305, 108766, 133191, 163099, 198697, 242069, 293535, 355788, 429609, 518396
OFFSET
0,3
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 1st equation.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/24) * (eta(q^2)^3 + 9 * eta(q^18)^3) / (eta(q) * eta(q^6)) in powers of q.
Expansion of phi(x) + 2*phi_{-}(x) in powers of x where phi() and phi_{-}() are 6th-order mock theta functions. [Ramanujan]
a(n) = A053268(n) + 2*A153251(n). [Ramanujan]
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(3/2)*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019
EXAMPLE
G.f. = 1 + x + 8*x^2 + 9*x^3 + 17*x^4 + 25*x^5 + 47*x^6 + 63*x^7 + ...
G.f. = 1/q + q^23 + 8*q^47 + 9*q^71 + 17*q^95 + 25*q^119 + 47*q^143 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 + 9 x^2 QPochhammer[ x^18]^3) / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^3*(1 - x^k)^2/(1 - x^(6*k)), {k, 1, nmax}] + 9*x^2*Product[(1 - x^(18*k))^3/((1 - x^k)*(1 - x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 + 9 * x^2 * eta(x^18 + A)^3) / (eta(x + A) * eta(x^6 + A)), n))};
CROSSREFS
Sequence in context: A350075 A353057 A274406 * A022098 A129659 A041130
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 18 2015
STATUS
approved