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 A232358 Expansion of phi(q^2)^2 / (phi(q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function. 4
 1, -2, 8, -16, 32, -60, 96, -160, 256, -394, 624, -944, 1408, -2092, 3008, -4320, 6144, -8612, 12072, -16720, 22976, -31424, 42528, -57312, 76800, -102254, 135728, -179104, 235264, -307852, 400704, -519808, 671744, -864672, 1109904, -1419456, 1809568 (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). LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (eta(q) * eta(q^16) * eta(q^4)^7)^2 / (eta(q^2 ) * eta(q^8))^9 in powers of q. Euler transform of period 16 sequence [-2, 7, -2, -7, -2, 7, -2, 2, -2, 7, -2, -7, -2, 7, -2, 0, ...]. G.f.: Product_{k>0} (1 + x^(4*k - 2))^5 / ((1 + x^(2*k - 1)) * (1 + x^(8*k - 4)))^2. a(n) = (-1)^n * A212318(n). a(2*n) = A014969(n). a(n) ~ (-1)^n * exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017 EXAMPLE G.f. = 1 - 2*q + 8*q^2 - 16*q^3 + 32*q^4 - 60*q^5 + 96*q^6 - 160*q^7 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[ -q^2, q^4]^5 / (QPochhammer[ -q, q^2] QPochhammer[ -q^4, q^8])^2, {q, 0, n}]; PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A) * eta(x^4 + A)^7)^2 / (eta(x^2 + A) * eta(x^8 + A))^9, n))}; CROSSREFS Cf. A014969, A212318. Sequence in context: A294553 A295949 A077666 * A212318 A346461 A232392 Adjacent sequences:  A232355 A232356 A232357 * A232359 A232360 A232361 KEYWORD sign AUTHOR Michael Somos, Nov 23 2013 STATUS approved

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Last modified May 20 08:03 EDT 2022. Contains 353852 sequences. (Running on oeis4.)