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 A233698 Expansion of b(q^2) * c(q^2) / (3 * b(q)^2) in powers of q where b(), c() are cubic AGM functions. 6
 1, 6, 25, 84, 248, 666, 1662, 3912, 8774, 18894, 39289, 79248, 155612, 298338, 559812, 1030224, 1862647, 3313494, 5807096, 10037796, 17129888, 28886052, 48170178, 79492824, 129900206, 210314976, 337545438, 537278124, 848509124, 1330069554, 2070183912 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 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. FORMULA Expansion of (eta(q^2) * eta(q^3) * eta(q^6) / eta(q)^3)^2 in powers of q. Euler transform of period 6 sequence [ 6, 4, 4, 4, 6, 0, ...]. a(n) = (-1)^n * A164271(n). 2 * a(n) = A132977(2*n + 1). -3 * a(n) =  A233670(6*n + 4). a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015 EXAMPLE G.f. = 1 + 6*x + 25*x^2 + 84*x^3 + 248*x^4 + 666*x^5 + 1662*x^6 + 3912*x^7 + ... G.f. = q^2 + 6*q^5 + 25*q^8 + 84*q^11 + 248*q^14 + 666*q^17 + 1662*q^20 + ... MATHEMATICA nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-2/3) *(eta[q^2]*eta[q^3]*eta[q^6]/eta[q]^3)^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 07 2018 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^3)^2, n))} CROSSREFS Cf. A132977, A164271, A233670. Sequence in context: A256859 A133714 A164271 * A230723 A220275 A055585 Adjacent sequences:  A233695 A233696 A233697 * A233699 A233700 A233701 KEYWORD nonn AUTHOR Michael Somos, Dec 14 2013 STATUS approved

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Last modified May 24 09:41 EDT 2019. Contains 323529 sequences. (Running on oeis4.)