A258096 Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.

1, 2, 4, 8, 7, 10, 12, 8, 18, 18, 16, 24, 21, 20, 28, 32, 20, 32, 36, 24, 42, 42, 28, 48, 57, 36, 52, 40, 36, 58, 60, 56, 48, 66, 48, 72, 74, 42, 80, 80, 61, 82, 72, 56, 90, 96, 64, 72, 98, 70, 100, 104, 64, 106, 108, 72, 114, 96, 84, 144, 111, 84, 104, 128

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * eta(q^2) * eta(q^4)^7 / (eta(q)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ 2, 1, 2, -6, 2, 1, 2, -4, ...].

a(n) = (-1)^n * A209940(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 7*x^4 + 10*x^5 + 12*x^6 + 8*x^7 + ...
G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 7*q^9 + 10*q^11 + 12*q^13 + 8*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4]^7 / (QPochhammer[ x]^2 QPochhammer[ x^8]^2), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] EllipticTheta[ 4, 0, x^4]^4 / (EllipticTheta[ 4, 0, x] 2 x^(1/2)), {x, 0, n}];

Prog

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

Crossrefs

Cf. A113417, A113419, A209940.

Sequence in context: A113417 A113419 A209940 * A261298 A133007 A141532

Adjacent sequences: A258093 A258094 A258095 * A258097 A258098 A258099

Keyword

nonn

Author

Michael Somos, May 19 2015

Status

approved