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 A262984 Expansion of f(-x^2, -x^10) / f(-x, -x) in powers of x where f(, ) is Ramanujan's general theta function. 1
 1, 2, 3, 6, 10, 16, 26, 40, 60, 90, 131, 188, 268, 376, 522, 720, 983, 1330, 1790, 2390, 3170, 4184, 5488, 7160, 9300, 12020, 15466, 19822, 25300, 32168, 40760, 51464, 64763, 81250, 101620, 126726, 157604, 195472, 241810, 298400, 367340, 451156, 552867, 676030 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 7th equation. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of psi(x^6) * phi(-x^2) / f(-x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions. Expansion of q^(-2/3) * eta(q^2)^2 * eta(q^12)^2 / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q. Euler transform of period 12 sequence [ 2, 0, 2, 1, 2, 1, 2, 1, 2, 0, 2, 0, ...]. -2 * a(n) = A262967(3*n + 2). a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015 G.f.: Sum_{k>=0} x^k * (Product_{i=1..k} 1 + x^(2*i)) / Product_{i=1..2*k+1} 1 - x^i). [Ramanujan] - Michael Somos, Nov 18 2015 EXAMPLE G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 26*x^6 + 40*x^7 + ... G.f. = q^2 + 2*q^5 + 3*q^8 + 6*q^11 + 10*q^14 + 16*q^17 + 26*q^20 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^2] / (2 x^(3/4) QPochhammer[ x]^2), {x, 0, n}]; nmax=60; CoefficientList[Series[Product[(1-x^(12*k)) * (1+x^(6*k)) * (1+x^(2*k-1)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))}; (PARI) q='q+O('q^99); Vec(eta(q^2)^2*eta(q^12)^2/(eta(q)^2*eta(q^4)*eta(q^6))) \\ Altug Alkan, Mar 19 2018 CROSSREFS Cf. A262967. Sequence in context: A280908 A146163 A101277 * A201077 A355383 A023655 Adjacent sequences: A262981 A262982 A262983 * A262985 A262986 A262987 KEYWORD nonn AUTHOR Michael Somos, Oct 06 2015 STATUS approved

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Last modified September 16 08:55 EDT 2024. Contains 375959 sequences. (Running on oeis4.)