A230057 Expansion of (3 * phi(q^3)^4 - phi(q)^4) / 2 in powers of q where phi () is a Ramanujan theta function.

1, -4, -12, -4, -12, -24, -12, -32, -12, -4, -72, -48, -12, -56, -96, -24, -12, -72, -12, -80, -72, -32, -144, -96, -12, -124, -168, -4, -96, -120, -72, -128, -12, -48, -216, -192, -12, -152, -240, -56, -72, -168, -96, -176, -144, -24, -288, -192, -12, -228

Offset

0,2

Comments

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).

Links

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

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of phi(q) * phi(q^3) * a(-q) in powers of q where a() is a cubic AGM theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = -12 * (t/i)^2 f(t) where q = exp(2 Pi i t).

a(n) = -4 * b(n) where b(n) is multiplicative and b(2^e) = 3 if e>0, b(3^e) = 1, b(p^e) = (p^(e+1) - 1)/(p - 1), if p>3. - Michael Somos, Nov 19 2013

G.f.: 1 - 4 * ( 1*x^1 / (1 + (-x)^1) + 2*x^2 / (1 + (-x)^2) + 4*x^4 / (1 + (-x)^4) + 5*x^5 / (1 + (-x)^5) + 7*x^7 / (1 + (-x)^7) + ...).

a(2*n) = a(4*n) = A186100(n). a(3*n) = a(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/2) * Pi * Gamma(11/12)^2 * Gamma(7/12)^2 / Gamma(3/4)^8 = A388813. - Simon Plouffe, Sep 18 2025

Sum_{k=0..n} a(k) ~ -c * n^2, where c = Pi^2/6 = 1.644934... (A013661). - Amiram Eldar, Dec 14 2025

Example

G.f. = 1 - 4*q - 12*q^2 - 4*q^3 - 12*q^4 - 24*q^5 - 12*q^6 - 32*q^7 + ...

Mathematica

a[ n_]:= SeriesCoefficient[(3*EllipticTheta[3, 0, q^3]^4 - EllipticTheta[ 3, 0, q]^4)/2, {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d * (2 + (-1)^(n/d)) * kronecker( 36, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (3 * sum( k=1, sqrtint(n\3), 2 * x^(3*k^2), 1 + A)^4 - sum( k=1, sqrtint(n), 2 * x^k^2, 1 + A)^4) / 2, n))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 1, (p^(e+1) - 1)/(p - 1))))))} /* Michael Somos, Nov 19 2013 */

Crossrefs

Cf. A013661, A186100, A388813.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A213343 A308518 A307874 * A205124 A224512 A010296

Adjacent sequences: A230054 A230055 A230056 * A230058 A230059 A230060

Keyword

sign,easy

Author

Michael Somos, Oct 07 2013

Status

approved