A263433 -id:A263433 - cp's OEIS frontend

A263527 Expansion of phi(-x^3) * f(-x^6)^3 / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

1, 0, 1, -2, 2, -2, 0, -4, 2, 0, 1, -4, 4, -2, 2, -4, 5, 0, 2, -2, 6, -4, 2, -4, 6, 0, 0, -6, 4, -2, 4, -8, 7, 0, 2, -10, 4, -6, 0, -4, 6, 0, 1, -6, 8, -6, 4, -8, 4, 0, 4, -8, 10, -4, 2, -8, 8, 0, 2, -6, 12, -4, 4, -8, 8, 0, 5, -8, 6, -4, 0, -8, 14, 0, 2, -10

Offset: 0

Michael Somos, Oct 19 2015

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

			G.f. = 1 + x^2 - 2*x^3 + 2*x^4 - 2*x^5 - 4*x^7 + 2*x^8 + x^10 - 4*x^11 + ...
G.f. = q^2 + q^8 - 2*q^11 + 2*q^14 - 2*q^17 - 4*q^23 + 2*q^26 + q^32 + ...

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. A261426, A261444, A263433, A263501.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^6]^3 / QPochhammer[ x^2], {x, 0, n}];

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

Expansion of q^(-2/3) * eta(q^3)^2 * eta(q^6)^2 / eta(q^2) in powers of q.
Euler transform of period 6 sequence [ 0, 1, -2, 1, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (2048/3)^(1/2) (t/I)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263501.
a(n) = (-1)^n * A261444(n). a(8*n + 1) = 0.
a(2*n) = A261426(n). a(4*n) = A263433(n). a(4*n + 2) = A261444(n).

A263444 Expansion of psi(q)^2 * chi(-q^6)^2 * f(-q^6) in powers of q where psi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 2, 2, 0, 0, -4, -3, -4, -4, 2, -6, 0, 2, 0, 2, -6, 4, -4, 0, 0, -8, 4, 0, 14, 2, 2, 12, 0, 8, -4, -3, 0, -4, 4, -4, 0, 4, 8, 12, -6, 0, -12, -6, 0, -8, 4, -6, 14, 5, 0, 0, 0, 0, -8, -6, -24, -12, 2, 0, 0, 6, 8, 2, -12, 8, -12, -6, 0, -8, 4, -12, 24, 6

Offset: 0

Michael Somos, Oct 18 2015

sign

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

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 - 4*x^7 - 3*x^8 - 4*x^9 - 4*x^10 + ...

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

Cf. A263433.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^2 QPochhammer[ q^6, q^12]^2 QPochhammer[ q^6] / (4 q^(1/4)), {q, 0, n}];

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

Expansion of psi(q)^2 * phi(-q^3) * chi(q^3)^2 in powers of q where psi(), phi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^6)^3 / (eta(q)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 2, -2, 2, -5, 2, -2, 2, -2, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263433.
a(8*n + 5) = a(9*n + 6) = 0.

cp's OEIS Frontend

A263527 Expansion of phi(-x^3) * f(-x^6)^3 / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

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