A131796 Expansion of chi(-q^3)^2 * chi(-q^5)^2 / (chi(-q) * chi(-q^15)) in powers of q where chi() is a Ramanujan theta function.

1, 1, 1, 0, 0, -1, -1, 0, 1, 0, -1, 0, 0, 1, 2, 1, -2, -3, -1, 1, 2, 3, 0, -3, -1, 2, 2, 0, -2, -6, -3, 4, 7, 3, -2, -5, -6, 2, 8, 3, -5, -6, -2, 4, 12, 7, -10, -15, -6, 5, 13, 12, -4, -18, -7, 11, 14, 6, -10, -24, -14, 20, 32, 12, -12, -29, -24, 9, 36, 15, -22, -30, -13, 22, 50, 27, -36, -63, -26, 24, 56, 45, -22, -69, -30, 42, 62

Offset

0,15

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

Euler transform of period 30 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 + u*(2 - 4*v + u*v).

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(3*k)) * (1 + x^(5*k)))^2.

a(n) = A131794(n) = -A131797(n) unless n=0.

Example

G.f. = 1 + q + q^2 - q^5 - q^6 + q^8 - q^10 + q^13 + 2*q^14 + q^15 - 2*q^16 - ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6] QPochhammer[ q^5, q^10])^2 / (QPochhammer[ q, q^2] QPochhammer[ q^15, q^30]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^15 + A)) * (eta(x^3 + A) * eta(x^5 + A) / (eta(x^6 + A) * eta(x^10 + A)))^2, n))};

Crossrefs

Cf. A131794, A131797.

Sequence in context: A145782 A131797 A145727 * A131794 A145726 A322984

Adjacent sequences: A131793 A131794 A131795 * A131797 A131798 A131799

Keyword

sign

Author

Michael Somos, Jul 16 2006

Status

approved