A164271 Expansion of ( f(-q^2) * f(q^3) * f(-q^6) / f(q)^3 )^2 in powers of q where f() is a Ramanujan theta function.

1, -6, 25, -84, 248, -666, 1662, -3912, 8774, -18894, 39289, -79248, 155612, -298338, 559812, -1030224, 1862647, -3313494, 5807096, -10037796, 17129888, -28886052, 48170178, -79492824, 129900206, -210314976, 337545438, -537278124, 848509124

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

Euler transform of period 12 sequence [ -6, 10, -4, 4, -6, 4, -6, 4, -4, 10, -6, 0, ...].

G.f.: (Product_{k>0} (1 - x^k + x^(2*k))^3 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k))^2 / ((1 + x^k)^2 * (1 - x^(2*k) + x^(4*k))))^2.

2 * a(n) = A164269(3*n + 2).

Expansion of ( f(-x^6)^6 / (f(x, x^5) * f(x)^2) )^2 in powers of x where f(,) is Ramanujan's general theta function. - Michael Somos, Sep 02 2015

Example

G.f. = 1 - 6*x + 25*x^2 - 84*x^3 + 248*x^4 - 666*x^5 + 1662*x^6 - 3912*x^8 + ...
G.f. = q^2 - 6*q^5 + 25*q^8 - 84*q^11 + 248*q^14 - 666*q^17 + 1662*q^20 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164271[n_] := SeriesCoefficient[((f[-q^2, -q^4]*f[q^3, -q^6]*f[-q^6, -q^12])/f[q, -q^2]^3)^2, {q, 0, n}]; Table[A164271[n], {n, 0, 50}] (* G. C. Greubel, Sep 16 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^4 / (eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)))^2, n))};

Crossrefs

Cf. A164269.

Sequence in context: A354396 A256859 A133714 * A233698 A230723 A220275

Adjacent sequences: A164268 A164269 A164270 * A164272 A164273 A164274

Keyword

sign

Author

Michael Somos, Aug 11 2009

Status

approved