A231948 -id:A231948 - cp's OEIS frontend

A231947 Expansion of q^(-1/3) * a(q)^2 * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions.

1, 13, 50, 72, 170, 205, 362, 360, 601, 650, 962, 864, 1370, 1224, 1850, 1584, 2451, 2210, 2880, 2520, 3722, 3277, 4490, 3600, 5330, 4706, 6242, 5040, 6912, 6120, 8500, 6624, 9410, 7813, 10610, 8424, 11882, 10250, 12672, 10440, 14521, 12506, 16130, 12240

Offset: 0

Michael Somos, Nov 15 2013

nonn

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

			G.f. = 1 + 13*x + 50*x^2 + 72*x^3 + 170*x^4 + 205*x^5 + 362*x^6 + 360*x^7 + ...
G.f. = q + 13*q^4 + 50*q^7 + 72*q^10 + 170*q^13 + 205*q^16 + 362*q^19 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A109041, A231948.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)*(eta[q]^3 + 9*eta[q^9]^3)^2*eta[q^3]/eta[q], {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3)^2 * eta(x^3 + A) / eta(x + A), n))}

Expansion of q^(-1/3) * (eta(q)^3 + 9 * eta(q^9)^3)^2 * eta(q^3) / eta(q) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231948.
-9 * a(n) = A109041(3*n + 1).

A231961 Expansion of b(q)^3 - 3*c(q)^3 in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -90, -216, -738, -1170, -1728, -2160, -4500, -3672, -6570, -6480, -8640, -9594, -15300, -10800, -17280, -18450, -20736, -19656, -32580, -22464, -36900, -32400, -38016, -36720, -54090, -36720, -59058, -58500, -60480, -53136, -86580, -58968, -86400, -77760

Offset: 0

Michael Somos, Nov 15 2013

sign

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

			G.f. = 1 - 90*q - 216*q^2 - 738*q^3 - 1170*q^4 - 1728*q^5 - 2160*q^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A231948, A231962.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3/ eta[q^3])^3 - 81*(eta[q^3]^3/eta[q])^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 - 81 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))};

Expansion of (eta(q)^3 / eta(q^3))^3 - 81 * (eta(q^3)^3 / eta(q))^3 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = - 3^(5/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231962.
a(n) = A231948(3*n) = A231962(3*n).

cp's OEIS Frontend

A231947 Expansion of q^(-1/3) * a(q)^2 * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions.

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