A271722 Expansion of phi(-x^3)^2 / (psi(-x) * psi(-x^3)) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 1, 1, -1, 0, 1, 0, -1, -1, 2, 0, -3, 1, 3, 1, -4, 0, 6, -2, -7, -1, 8, 0, -10, 2, 13, 2, -16, 0, 18, -2, -22, -3, 28, 0, -33, 3, 38, 3, -45, 0, 55, -4, -65, -4, 74, 0, -87, 5, 104, 6, -121, 0, 138, -6, -160, -7, 188, 0, -217, 7, 247, 8, -284, 0, 330, -10

Offset

0,10

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 psi(x^2) / psi(x^6) + x * psi(x^6) / psi(x^2) in powers of

Expansion of phi(-x^6) * chi(-x^3)^3 / psi(-x) = psi(-x^3)^3 / (psi(-x) * psi(x^6)^2) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of q^(1/2) * eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 1, 0, -2, 1, 1, -2, 1, 1, -2, 0, 1, 0, ...].

a(2*n) = A256626(n). a(2*n + 1) = A101195(n).

Example

G.f. = 1 + x + x^2 - x^3 + x^5 - x^7 - x^8 + 2*x^9 - 3*x^11 + x^12 + ...
G.f. = q^-1 + q + q^3 - q^5 + q^9 - q^13 - q^15 + 2*q^17 - 3*q^21 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2 x^(1/2) EllipticTheta[ 4, 0, x^3]^2 / (EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)]), {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^3 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A)), n))};

Crossrefs

Sequence in context: A324115 A029239 A162201 * A029219 A339373 A212217

Adjacent sequences: A271719 A271720 A271721 * A271723 A271724 A271725

Keyword

sign

Author

Michael Somos, Apr 12 2016

Status

approved