A257399 Expansion of phi(x^3) * phi(-x^12) / chi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0

Offset

0,4

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 (phi(-x^24)^2 + 2 * x^3 * psi(-x^12)^2) / chi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(-1/6) * eta(q^6)^5 * eta(q^8) / (eta(q^4) * eta(q^3)^2 * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 0, 0, 2, 1, 0, -3, 0, 0, 2, 0, 0, -2, 0, 0, 2, 0, 0, -3, 0, 1, 2, 0, 0, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257400.

a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A257398(n). a(4*n + 3) = 2 * A255317(n).

Example

G.f. = 1 + 2*x^3 + x^4 + 2*x^7 + x^8 + 2*x^11 + 2*x^12 + 2*x^16 + 3*x^20 + ...
G.f. = q + 2*q^19 + q^25 + 2*q^43 + q^49 + 2*q^67 + 2*q^73 + 2*q^97 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^12], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^8 + A) / (eta(x^4 + A) * eta(x^3 + A)^2 * eta(x^24 + A)), n))};

Crossrefs

Cf. A204531, A255317, A257398, A257400.

Sequence in context: A262929 A226862 A226864 * A168313 A072575 A025872

Adjacent sequences: A257396 A257397 A257398 * A257400 A257401 A257402

Keyword

nonn

Author

Michael Somos, Apr 21 2015

Status

approved