A213056 Expansion of chi(x) * f(x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.

1, 1, 0, 4, 4, 1, 4, 4, 5, 0, 0, 8, 4, 4, 4, 8, 9, 4, 0, 4, 12, 1, 4, 8, 8, 4, 0, 8, 8, 4, 8, 16, 8, 5, 0, 12, 12, 0, 8, 12, 13, 0, 0, 8, 8, 8, 12, 8, 16, 4, 0, 16, 12, 4, 4, 20, 13, 4, 0, 16, 20, 8, 8, 8, 8, 9, 0, 12, 16, 4, 12, 12, 16, 0, 0, 16, 20, 4, 8

Offset

0,4

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^6)^9 / (eta(q) * eta(q^3)^3 * eta(q^4) * eta(q^12)^3) in powers of q.

Expansion of q^(-1/9) times theta series of cubic lattice with respect to point [0, 0, 1/3] in powers of q^(1/3).

Euler transform of period 12 sequence [ 1, -1, 4, 0, 1, -7, 1, 0, 4, -1, 1, -3, ...].

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

a(4*n + 1) = a(n). a(8*n + 2) = 0.

Example

G.f. = 1 + x + 4*x^3 + 4*x^4 + x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 8*x^11 + 4*x^12 + ...
G.f. = q + q^4 + 4*q^10 + 4*q^13 + q^16 + 4*q^19 + 4*q^22 + 5*q^25 + 8*q^34 + ...

Mathematica

CoefficientList[QPochhammer[q^2]^2*QPochhammer[-q^3]^3 / (QPochhammer[q] * QPochhammer[q^4]) + O[q]^80, q] (* Jean-François Alcover, Nov 05 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)* eta[q^2]^2*eta[q^6]^9/(eta[q]*eta[q^3]^3*eta[q^4]*eta[q^12]^3), {q, 0, 50}], q] (* G. C. Greubel, Aug 12 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^9 / (eta(x + A) * eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A)^3) , n))}

Crossrefs

Sequence in context: A087709 A106642 A327855 * A135012 A156380 A329708

Adjacent sequences: A213053 A213054 A213055 * A213057 A213058 A213059

Keyword

nonn

Author

Michael Somos, Jun 03 2012

Status

approved