A215472 Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -14, 81, -238, 322, 0, -429, 82, 0, 2162, -3038, -1134, 2401, 2482, 0, -6958, 3332, 0, 1442, 0, 6561, -4508, -9758, 0, -1918, 18802, 0, 9362, -24638, -19278, 14641, 14756, 0, 0, 6562, 0, -1148, -33998, 26082, -20398, 0, 0, 28083, 49042, 0, -64078, -30268

Offset

0,2

Comments

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

This is a member of an infinite family of integer weight level 8 modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

Links

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

Shobhit, How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/4) * eta(q)^14 / eta(q^2)^4 in powers of q.

Expansion of q^(-1/4) * ( eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 + 4 * eta(q^2)^4 * eta(q^4)^2 * eta(q^8)^4 ) in powers of q. - Michael Somos, Sep 05 2013

Euler transform of period 2 sequence [ -14, -10, ...].

a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 128 (t/i)^5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030212.

a(n) = (-1)^n * A209942(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n).

a(n) = A030212(4*n + 1). - Michael Somos, Sep 05 2013

Example

1 - 14*x + 81*x^2 - 238*x^3 + 322*x^4 - 429*x^6 + 82*x^7 + 2162*x^9 + ...
q - 14*q^5 + 81*q^9 - 238*q^13 + 322*q^17 - 429*q^25 + 82*q^29 + 2162*q^37 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^14 / QPochhammer[ x^2]^4, {x, 0, n}] (* Michael Somos, Sep 05 2013 *)

Prog

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

Crossrefs

Cf. A000729, A002171, A008441, A030212, A209942, A215601.

Sequence in context: A329820 A239421 A309338 * A209942 A215700 A199912

Adjacent sequences: A215469 A215470 A215471 * A215473 A215474 A215475

Keyword

sign

Author

Michael Somos, Aug 12 2012

Status

approved