A282031 -id:A282031 - cp's OEIS frontend

A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

1, -12, 60, -156, 204, -72, -84, -96, 492, -588, 360, -144, 60, -168, 480, -936, 1068, -216, -516, -240, 1224, -1248, 720, -288, 348, -372, 840, -1884, 1632, -360, -504, -384, 2220, -1872, 1080, -576, -372, -456, 1200, -2184, 2952, -504, -672, -528, 2448

Offset: 0

Michael Somos, Sep 26 2013

sign

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).

			G.f. = 1 - 12*q + 60*q^2 - 156*q^3 + 204*q^4 - 72*q^5 - 84*q^6 - 96*q^7 + ...

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
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Cf. A122859, A229615.

A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] - 12*A[2] + 60*A[3];

a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -7, 10, -7, 1, 2}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -3, 4, -3, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^6 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}];

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, n/d * [2, 1, -7, 10, -7, 1][d%6 + 1]))};

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * [0, 1, -3, 4, -3, 1][d%6 + 1]))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2))^2, n))};

A = ModularForms( Gamma0(6), 2, prec=50).basis(); A[0] - 12*A[1] + 60*A[2];

Expansion of (2*a(q^2) - a(q))^2 = b(q)^4 / b(q^2)^2 in powers of q where a(), b() are cubic AGM theta functions.
Expansion of (eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2))^2 in powers of q.
Euler transform of period 6 sequence [-12, -6, -8, -6, -12, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229615.
G.f.: ( Product_{k>0} (1 + x^(3*k)) * (1 - x^k)^3 / ((1 + x^k)^3 * (1 - x^(3*k))))^2.
Convolution square of A122859.
Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

A122858 Expansion of E(k) * K(k) * (2/Pi)^2 in powers of q^2 where E(), K() are complete elliptic integrals and the nome q = exp( -Pi * K(k') / K(k)).

Original entry on oeis.org

1, 8, -8, 32, -40, 48, -32, 64, -104, 104, -48, 96, -160, 112, -64, 192, -232, 144, -104, 160, -240, 256, -96, 192, -416, 248, -112, 320, -320, 240, -192, 256, -488, 384, -144, 384, -520, 304, -160, 448, -624, 336, -256, 352, -480, 624, -192, 384, -928, 456

Offset: 0

Michael Somos, Sep 15 2006

sign

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 + 8*q - 8*q^2 + 32*q^3 - 40*q^4 + 48*q^5 - 32*q^6 + 64*q^7 - 104*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
A. Brini and A. Tanzini, Exact results for topological strings on resolved Y(p,q) singularities, arXiv:0804.2598 [hep-th], 2008-2009, p. 40, equation (6.82).

Cf. A002129, A143336.

a[n_] := SeriesCoefficient[8 q D[Series[EllipticTheta[2, 0, q^(1/2)], {q, 0, n + 1}], q] / Series[EllipticTheta[2, 0, q^(1/2)], {q, 0, n + 1}], {q, 0, n}] (* Sander Mack-Crane, Nov 07 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -8 DivisorSum[ n, # (-1)^# &]]; (* Michael Somos, Jun 02 2015 *)
a[ n_] := SeriesCoefficient[ With[{f = EllipticTheta[ 2, 0, q^(1/2)]}, 8 q D[f + O[q]^(n + 1), q] / f], {q, 0, n}]; (* Michael Somos, Jun 02 2015 *)
CoefficientList[Series[(2/Pi) EllipticE[InverseEllipticNomeQ[Sqrt[q]]] EllipticTheta[3, 0, Sqrt[q]]^2, {q, 0, 40}], q] (* Jan Mangaldan, Jul 07 2020 *)

{a(n) = if( n<1, n==0, -8 * sumdiv(n, d, (-1)^d * d))};

Expansion of (2 * E(k) - k'^2 * K(k)) * K(k) * (2/Pi)^2 in powers of q.
Expansion of (E(k) + k' * K(k)) * K(k) * (2/Pi)^2 / 2 in powers of q^4.
Expansion of (4 * P(q^2) - P(q)) / 3 in powers of q where P() is a Ramanujan Lambert series.
G.f.: 1 + 8 * Sum_{k>0} x^k / (1 + x^k)^2.
G.f.: 1 - 8 * Sum_{k>0} k * (-x)^k / (1 - x^k).
G.f.: 1 + 8 * Sum_{k>0} k * x^k * (1 - 3*x^k) / (1 - x^(2*k)).
a(n) = 8 * A002129(n) unless n=0. a(n) = (-1)^n * A143336(n).
Expansion of 8*q*theta_2(0,q)' / theta_2(0,q) in powers of q=exp(2*Pi*i*tau), where theta_2(z,q) is a Jacobi theta function. - Sander Mack-Crane, Nov 07 2013
Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

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A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

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A122858 Expansion of E(k) * K(k) * (2/Pi)^2 in powers of q^2 where E(), K() are complete elliptic integrals and the nome q = exp( -Pi * K(k') / K(k)).

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