A227595 Expansion of phi(-x) * psi(x^3)^2 / chi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, -2, 0, 3, -4, 0, 4, -2, 0, 5, -6, 0, 5, -8, 0, 5, -2, 0, 7, -10, 0, 7, -8, 0, 9, 0, 0, 7, -12, 0, 6, -12, 0, 11, -6, 0, 8, -10, 0, 10, -12, 0, 8, -4, 0, 9, -12, 0, 14, -16, 0, 10, 0, 0, 15, -14, 0, 7, -16, 0, 7, -8, 0, 14, -18, 0, 14, -12, 0, 16, -2, 0, 8

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^(-7/8) * eta(q)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^3) in powers of q.

Euler transform of period 6 sequence [ -2, -1, 1, -1, -2, -3, ...].

a(3*n + 2) = a(27*n + 25) = 0.

Example

1 - 2*x + 3*x^3 - 4*x^4 + 4*x^6 - 2*x^7 + 5*x^9 - 6*x^10 + 5*x^12 - 8*x^13 + ...
q^7 - 2*q^15 + 3*q^31 - 4*q^39 + 4*q^55 - 2*q^63 + 5*q^79 - 6*q^87 + 5*q^103 + ...

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -x]*EllipticTheta[2, 0, x^(3/2)]^2/(4*x^(3/4)*QPochhammer[x^3, x^6]), {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 08 2017 *)

Prog

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

Crossrefs

Sequence in context: A153250 A102389 A099091 * A078436 A368090 A209705

Adjacent sequences: A227592 A227593 A227594 * A227596 A227597 A227598

Keyword

sign

Author

Michael Somos, Jul 17 2013

Status

approved