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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 (list; graph; refs; listen; history; text; internal format)
 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 REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 1st equation.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 G. C. Greubel, Table of n, a(n) for n = 0..1000 Michael Somos, Introduction to Ramanujan theta functions 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 Cf. A053268, A153251. Sequence in context: A350075 A353057 A274406 * A022098 A129659 A041130 Adjacent sequences: A261451 A261452 A261453 * A261455 A261456 A261457 KEYWORD nonn AUTHOR Michael Somos, Nov 18 2015 STATUS approved

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Last modified September 29 23:48 EDT 2023. Contains 365781 sequences. (Running on oeis4.)