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A123629
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Expansion of b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.
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6
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1, 3, 6, 11, 18, 30, 48, 75, 114, 170, 252, 366, 526, 744, 1044, 1451, 1998, 2730, 3700, 4986, 6672, 8876, 11736, 15438, 20207, 26322, 34134, 44072, 56682, 72612, 92680, 117867, 149400, 188758, 237744, 298554, 373838, 466836, 581412, 722266, 895014
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OFFSET
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1,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
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FORMULA
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Expansion of (eta(q^3) / eta(q^6))^2 * (eta(q^2) * eta(q^18) / (eta(q) * eta(q^9)))^3 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - u*(6*v + 4*v^2).
Euler transform of period 18 sequence [ 3, 0, 1, 0, 3, 0, 3, 0, 4, 0, 3, 0, 3, 0, 1, 0, 3, 0, ...].
Convolution inverse is A123676. - Michael Somos, Feb 19 2015
Expansion of q * chi(-q^3)^2 / (chi(-q) * chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Feb 19 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123676. - Michael Somos, Feb 19 2015
a(3*n) = 6 * A128638(n). a(3*n + 2) = 3 * A233698(n). - Michael Somos, Feb 19 2015
a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 10 2015
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EXAMPLE
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G.f. = q + 3*q^2 + 6*q^3 + 11*q^4 + 18*q^5 + 30*q^6 + 48*q^7 + 75*q^8 + ...
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MATHEMATICA
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nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1+x^(9*k))^3 / (1+x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 10 2015 *)
A123629[n_] := SeriesCoefficient[q*(QPochhammer[q^3]/QPochhammer[q^6])^2*(QPochhammer[q^2]*QPochhammer[q^18]/(QPochhammer[q]*QPochhammer[q^9] ))^3, {q, 0, n}]; Table[A123629[n], {n, 0, 50}] (* G. C. Greubel, Oct 09 2017 *)
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^6 + A))^2 * (eta(x^2 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^9 + A)))^3, n))};
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CROSSREFS
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Cf. A123676, A128638, A233698.
Sequence in context: A147079 A281572 A152074 * A212484 A279100 A347415
Adjacent sequences: A123626 A123627 A123628 * A123630 A123631 A123632
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Oct 03 2006
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EXTENSIONS
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Typo in xrefs corrected by Vaclav Kotesovec, Oct 10 2015
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STATUS
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approved
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