A208604 -id:A208604 - cp's OEIS frontend

A208274 Expansion of phi(q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 0, -4, 0, 0, 0, 10, 0, 0, 0, -20, 0, 0, 0, 36, 0, 0, 0, -64, 0, 0, 0, 110, 0, 0, 0, -180, 0, 0, 0, 288, 0, 0, 0, -452, 0, 0, 0, 692, 0, 0, 0, -1044, 0, 0, 0, 1554, 0, 0, 0, -2276, 0, 0, 0, 3296, 0, 0, 0, -4724, 0, 0, 0, 6696, 0, 0, 0, -9408, 0, 0

Offset: 0

Michael Somos, Mar 12 2012

sign

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

Differs from A127391 only at n=0. - R. J. Mathar, Mar 18 2012

			1 + 2*q - 4*q^5 + 10*q^9 - 20*q^13 + 36*q^17 - 64*q^21 + 110*q^25 - 180*q^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A079006, A112128, A134746, A208604.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5), n))}

Expansion of eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -3, 2, -3, 2, 2, 2, -3, 2, -3, 2, -3, 2, 0, ...].
G.f. A(x) satisfies  A(x)^2 - 2*A(x) + 2 = A134746(x^2), which means (phi(q) / phi(q^4) - 1)^2 + 1 = (phi(q^2) / phi(q^4))^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - 2*u + 2) * (v^2 - 2*v + 2) - v^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2 - u) * v * (v - 1) * (2 - v) - (u - v)^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = g(t) where q = exp(2 Pi i t) and g() is g.f. for A112128.
a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A079006(n). a(n) = (-1)^n * A208604(n). Convolution inverse is A112128.

A208933 Expansion of phi(q^4) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 48, 80, 128, 202, 312, 472, 704, 1036, 1504, 2160, 3072, 4324, 6036, 8360, 11488, 15680, 21264, 28656, 38400, 51182, 67864, 89552, 117632, 153836, 200352, 259904, 335872, 432480, 554952, 709728, 904784, 1149916, 1457136, 1841200, 2320128

Offset: 0

Michael Somos, Mar 13 2012

nonn

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

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 16*q^4 + 28*q^5 + 48*q^6 + 80*q^7 + 128*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A093160, A112128, A123655, A131126, A208603, A208604.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Apr 25 2015 *)
nmax=60; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(8*k))^5 / ((1-x^k)^2 * (1-x^(4*k))^2 * (1-x^(16*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^16 + A))^2, n))};

Expansion of eta(q^2) * eta(q^8)^5 / (eta(q) * eta(q^4) * eta(q^16))^2 in powers of q.
Euler transform of period 16 sequence [ 2, 1, 2, 3, 2, 1, 2, -2, 2, 1, 2, 3, 2, 1, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A208603.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2*u - 1) * (2*v^2 - 2*v + 1) - u^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2*u - 1) * v * (v - 1) * (2*v - 1) - (u - v)^4.
(-1)^n * a(n) = A112128(n). a(n) = 2 * A123655(n) unless n=0. 2 * a(n) = A007096(n) unless n=0. a(2*n) = A131126(n). a(2*n + 1) = 2 * A093160(n). Convolution inverse of A208604.
G.f.: (Sum_{k in Z} x^(4 * k^2)) / (Sum_{k in Z} (-1)^k * x^(k^2)) = theta_3(x^4) / theta_3(-x).
G.f.: Product_{k>0} ((1 + x^(2*k)) * (1 + x^(4*k)))^3 / ((1 + (-x)^k) * (1 + x^(8*k)))^2.
a(n) ~ exp(sqrt(n)*Pi) / (2^(7/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

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A208274 Expansion of phi(q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

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