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A080015 Expansion of theta_3(q) / theta_3(q^2) in powers of q. 5
1, 2, -2, -4, 6, 8, -12, -16, 22, 30, -40, -52, 68, 88, -112, -144, 182, 228, -286, -356, 440, 544, -668, -816, 996, 1210, -1464, -1768, 2128, 2552, -3056, -3648, 4342, 5160, -6116, -7232, 8538, 10056, -11820, -13872, 16248, 18996, -22176, -25844, 30068 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 214 Entry 24(ii).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Expansion of eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^7) in powers of q.

Euler transform of period 8 sequence [ 2, -5, 2, 2, 2, -5, 2, 0, ...].

G.f.: A(x)/B(x), where A(x) = Sum_{m = -infinity..infinity} x^(m^2) and B(x) = Sum_{m = -infinity..infinity} x^(2*m^2). - Vladeta Jovovic, Mar 22 2005

Expansion of phi(x) / phi(x^2) where phi() is a Ramanujan theta function.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + (1 - u*v)^2 - v^2. - Michael Somos, Jan 31 2006

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^4 - v^4 + 8 * u*v - 6 * u*v * (u^2 + v^2) + 4 * (u*v)^3. - Michael Somos, Jan 31 2006

Expansion of sqrt(m) in powers of q where m is the multiplier for the second degree modular equation.

G.f.: Prod_{k>0} ((1 - x^(8*k - 2)) * (1 - x^(8*k - 6)))^5 / ((1 - x^(8*k - 1)) * (1 - x^(8*k - 3)) * (1 - x^(8*k - 4)) * (1 - x^(8*k - 5)) * (1 - x^(8*k - 7)))^2.

a(n) = (-1)^n * A210030(n). a(n) = (-1)^[n/2] * A080054(n).

EXAMPLE

G.f. = 1 + 2*q - 2*q^2 - 4*q^3 + 6*q^4 + 8*q^5 - 12*q^6 - 16*q^7 + 22*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)

PROG

(PARI) {a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + 2 * x + O(x^2); while( m<n, m*=2; A = subst(A, x, x^2); A = (1 + 2 * sqrt((A^2 - 1) / 4)) / A); polcoeff( A, n))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7, n))};

CROSSREFS

Cf. A080054, A210030.

Sequence in context: A320193 A260215 A261156 * A210030 A080054 A108494

Adjacent sequences:  A080012 A080013 A080014 * A080016 A080017 A080018

KEYWORD

sign,easy

AUTHOR

Michael Somos, Jan 20 2003

STATUS

approved

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Last modified June 1 13:20 EDT 2020. Contains 334762 sequences. (Running on oeis4.)