A132179 Expansion of f(-x^2)^2 * f(x, x^2) / f(-x^3)^3 in powers of x where f(,) is a Ramanujan theta function.

1, 1, -1, 1, 0, -3, 4, 1, -6, 5, 1, -10, 11, 4, -19, 17, 4, -31, 31, 9, -50, 46, 11, -79, 77, 21, -122, 112, 28, -183, 173, 46, -273, 249, 62, -396, 370, 98, -573, 521, 130, -815, 751, 193, -1149, 1041, 261, -1599, 1461, 373, -2214, 1998, 498, -3031, 2750, 696, -4125, 3708, 923, -5567

Offset

0,6

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 (chi(-x) / chi(-x^3)^3) * (psi(x) / psi(x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 05 2015

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

Euler transform of period 6 sequence [ 1, -2, 2, -2, 1, 0, ...].

Given g.f. A(x), then B(q) = A(q^6)/q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - 3*v)^3 - 4*(u^2*v^2 - v^3)*(u^2*v^2 - 2*v^3).

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132180.

Convolution of A092848 and A058487. - Michael Somos, Feb 05 2015

a(n) = (-1)^n * A254525(n) = A062242(2*n) = A062244(2*n) = A132301(2*n) = A182036(3*n). - Michael Somos, Feb 05 2015

a(2*n) = A230256(n). a(2*n + 1) = A233037(n). - Michael Somos, Feb 05 2015

Example

G.f. = 1 + x - x^2 + x^3 - 3*x^5 + 4*x^6 + x^7 - 6*x^8 + 5*x^9 + x^10 + ...
G.f. = 1/q + q^5 - q^11 + q^17 - 3*q^29 + 4*q^35 + q^41 - 6*q^47 + 5*q^53 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 / (QPochhammer[ x]  QPochhammer[ x^3] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Feb 05 2015 *)

Prog

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

Crossrefs

Cf. A058487, A062242, A062244, A092848, A132180, A132301, A182036, A230256, A233037, A254525.

Sequence in context: A030707 A349629 A254525 * A378210 A089029 A131226

Adjacent sequences: A132176 A132177 A132178 * A132180 A132181 A132182

Keyword

sign

Author

Michael Somos, Aug 12 2007

Status

approved