A187153 - OEIS

%I #27 Mar 30 2021 21:59:42

%S 1,1,-1,-3,-2,3,8,5,-7,-18,-12,15,38,24,-30,-75,-46,57,140,86,-104,

%T -252,-152,183,439,262,-313,-744,-442,522,1232,725,-852,-1998,-1168,

%U 1365,3182,1852,-2150,-4986,-2886,3336,7700,4436,-5106,-11736,-6736,7719

%N Expansion of q * (psi(q) / psi(q^2)) / (psi(q^3) / psi(q^6))^3 in powers of q where psi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C The Alaca, et. al. paper in equation (1.8) on page 178 uses the notation p(q) := (phi^2(q) - phi^2(q^3)/(2 phi^2(q^3)), where phi(q) is a Ramanujan theta function. This p(q) is twice the g.f. of this sequence. - _Michael Somos_, Mar 30 2021

%H G. C. Greubel, <a href="/A187153/b187153.txt">Table of n, a(n) for n = 1..1000</a>

%H A. Alaca, S. Alaca, K. S. Williams, <a href="https://people.math.carleton.ca/%7Eaalaca/pubs/aa2006-1.pdf">On the two-dimensional theta functions of the Borweins</a>

%H Johannes Blümlein, <a href="https://arxiv.org/abs/1808.08128">Iterative Non-iterative Integrals in Quantum Field Theory</a>, arXiv:1808.08128 [hep-th], 2018.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H W. Zudilin, <a href="http://mathoverflow.net/q/50804/">Deciding whether a given power series is modular or not</a>

%F Expansion of (eta(q^2) * eta(q^3) * eta(q^12)^2)^3 / (eta(q) * eta(q^4)^2 * eta(q^6)^9) in powers of q.

%F Euler transform of period 12 sequence [1, -2, -2, 0, 1, 4, 1, 0, -2, -2, 1, 0, ...].

%F Expansion of c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.

%F If p = 2 * A(q), then B(q) = p * ((2 + p) / (1 + 2*p))^3 and B(q^3) = p^3 * ((2 + p) / (1 + 2*p)) where B() is the g.f. for A115977. - _Michael Somos_, Feb 27 2012

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u * (1 + 2*v))^2 - v * (1 + 2*u).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^2 * (1 + 4*v) * (3*v + u*(1 + 4*v)) - v * (1 + v) * (3*u + 1 + v).

%F Convolution inverse of A187143.

%F Expansion of (phi^2(q) - phi^2(q^3)/(4*phi^2(q^3)), where phi(q) is a Ramanujan theta function. - _Michael Somos_, Mar 30 2021

%e G.f. = q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 - 18*q^10 + ...

%t QP = QPochhammer; s = (QP[q^2]*QP[q^3]*QP[q^12]^2)^3/(QP[q]*QP[q^4]^2* QP[q^6]^9) + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)

%t QP = QPochhammer; Rest[Table[SeriesCoefficient[q*(QP[-q, q^2]*QP[-q^6, q^6]^3)/(QP[-q^2, q^2]*QP[-q^3, q^6]^3), {q, 0, n}], {n, 0, 50}]] (* _G. C. Greubel_, Dec 04 2017 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^2, x^4]^2 QPochhammer[x^3, x^6]^3 QPochhammer[ -x^6, x^6]^6, {x, 0, n}]; (* _Michael Somos_, Oct 15 2018 *)

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, x]^2 / EllipticTheta[3, 0, x^3]^2 - 1)/4, {x, 0, n}]; (* _Michael Somos_, Mar 30 2021 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^2)^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^9), n))};

%Y Cf. A115977, A123649, A187143, A213265.

%K sign

%O 1,4

%A _Michael Somos_, Mar 06 2011