A232358 Expansion of phi(q^2)^2 / (phi(q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function.

1, -2, 8, -16, 32, -60, 96, -160, 256, -394, 624, -944, 1408, -2092, 3008, -4320, 6144, -8612, 12072, -16720, 22976, -31424, 42528, -57312, 76800, -102254, 135728, -179104, 235264, -307852, 400704, -519808, 671744, -864672, 1109904, -1419456, 1809568

Offset

0,2

Comments

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

Links

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

Formula

Expansion of (eta(q) * eta(q^16) * eta(q^4)^7)^2 / (eta(q^2 ) * eta(q^8))^9 in powers of q.

Euler transform of period 16 sequence [-2, 7, -2, -7, -2, 7, -2, 2, -2, 7, -2, -7, -2, 7, -2, 0, ...].

G.f.: Product_{k>0} (1 + x^(4*k - 2))^5 / ((1 + x^(2*k - 1)) * (1 + x^(8*k - 4)))^2.

a(n) = (-1)^n * A212318(n). a(2*n) = A014969(n).

a(n) ~ (-1)^n * exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = 1 - 2*q + 8*q^2 - 16*q^3 + 32*q^4 - 60*q^5 + 96*q^6 - 160*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^2, q^4]^5 / (QPochhammer[ -q, q^2] QPochhammer[ -q^4, q^8])^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A) * eta(x^4 + A)^7)^2 / (eta(x^2 + A) * eta(x^8 + A))^9, n))};

Crossrefs

Cf. A014969, A212318.

Sequence in context: A294553 A295949 A077666 * A212318 A346461 A232392

Adjacent sequences: A232355 A232356 A232357 * A232359 A232360 A232361

Keyword

sign

Author

Michael Somos, Nov 23 2013

Status

approved