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 A123861 Expansion of (f(q) * f(q^3) / (f(-q) * f(-q^3)))^2 in powers of q where f() is a Ramanujan theta function. 3
 1, 4, 8, 20, 48, 88, 168, 320, 544, 932, 1584, 2544, 4080, 6488, 9984, 15288, 23232, 34568, 51144, 75152, 108832, 156736, 224352, 317728, 447648, 627292, 871856, 1206068, 1660416, 2271032, 3092976, 4194464, 5657728, 7602096, 10175760 (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..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. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of phi(q) * phi(q^3) / (phi(-q) * phi(-q^3)) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Aug 31 2014 Expansion of eta(q^2)^6 * eta(q^6)^6 / (eta(q)^4 * eta(q^3)^4 * eta(q^4)^2 * eta(q^12)^2) in powers of q. - Michael Somos, Aug 31 2014 Euler transform of period 12 sequence [4, -2, 8, 0, 4, -4, 4, 0, 8, -2, 4, 0, ...]. G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u - 1)^2 - 4 * u*v * (v - 1). Let g.f. A(x) = u, then B(x) = u * (u-1) / 4, B(x^2) = ((u-1) / 4)^2 / u where B(x) is the g.f. for A123653. G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187197. - Michael Somos, Aug 31 2014 a(n) = 4 * A123647(n) unless n=0. a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015 EXAMPLE G.f. = 1 + 4*q + 8*q^2 + 20*q^3 + 48*q^4 + 88*q^5 + 168*q^6 + 320*q^7 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] QPochhammer[ -q^3] / (QPochhammer[ q] QPochhammer[ q^3]))^2, {q, 0, n}]; (* Michael Somos, Aug 31 2014 *) nmax=60; CoefficientList[Series[Product[(1+x^k)^6 * (1-x^k)^2 * (1+x^(3*k))^4 / ((1-x^(4*k))^2 * (1+x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^6 + A)^6 / (eta(x + A)^4 * eta(x^3 + A)^4 * eta(x^4 + A)^2 * eta(x^12 + A)^2), n))}; CROSSREFS Cf. A123647, A187197. Sequence in context: A190589 A009916 A203167 * A115099 A060919 A009333 Adjacent sequences:  A123858 A123859 A123860 * A123862 A123863 A123864 KEYWORD nonn AUTHOR Michael Somos, Oct 14 2006 STATUS approved

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Last modified November 23 17:30 EST 2020. Contains 338595 sequences. (Running on oeis4.)