A164268 Expansion of f(q^3) * phi(q^3)^3 / (q * f(q^9)^3 * phi(q)) in powers of q where f(), phi() are Ramanujan theta functions.

1, -2, 4, -1, 0, 4, 1, 0, 0, 1, 0, -8, -1, 0, -8, 0, 0, 4, 1, 0, 16, -2, 0, 16, 0, 0, -4, 2, 0, -32, -3, 0, -32, 1, 0, 8, 4, 0, 56, -4, 0, 56, 1, 0, -16, 4, 0, -96, -6, 0, -92, 1, 0, 24, 5, 0, 160, -8, 0, 152, 1, 0, -40, 8, 0, -252, -10, 0, -240, 2, 0, 64, 11

Offset

-1,2

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 = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of chi(q^9)^3 / (q * chi(q^3)) - 2 + 4 * q * chi(q^3) / chi(q^9)^3 in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 * eta(q^9)^3 * eta(q^36)^3 / (eta(q^2)^5 * eta(q^3)^7 * eta(q^12)^7 * eta(q^18)^9) in powers of q.

Euler transform of period 36 sequence [ -2, 3, 5, 1, -2, -8, -2, 1, 2, 3, -2, -3, -2, 3, 5, 1, -2, -2, -2, 1, 5, 3, -2, -3, -2, 3, 2, 1, -2, -8, -2, 1, 5, 3, -2, 0, ...].

a(3*n) = 0 unless n=0. a(3*n - 1) = A062244(n). a(3*n + 1) = 4 * A128111(n).

a(n) = A164612(n) unless n=0. a(6*n + 1) = 4 * A233034(n). a(6*n + 2) = - A092848(n). a(6*n + 4) = 4 * A216046(n). a(6*n + 5) = A132179(n). - Michael Somos, Sep 05 2015

a(12*n - 1) = A230256(n). a(12*n + 2) = - A233034(n). a(12*n + 5) = A233037(n). a(12*n + 8) = A216046(n). - Michael Somos, Sep 05 2015

Convolution inverse of A164269. - Michael Somos, Sep 05 2015

Example

G.f. = 1/q - 2 + 4*q - q^2 + 4*q^4 + q^5 + q^8 - 8*q^10 - q^11 - 8*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q^3] EllipticTheta[ 3, 0, q^3]^3 / (QPochhammer[ -q^9]^3 EllipticTheta[ 3, 0, q]), {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
a[ n_] := SeriesCoefficient[ With[ {A = 1/q QPochhammer[ -q^9, q^18]^3 QPochhammer[ q^3, -q^3]}, A - 2 + 4 / A], {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3 / (eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9), n))};

Crossrefs

Cf. A062244, A092848, A128111, A132179, A164269, A164612, A216046, A230256, A233034, A233037.

Sequence in context: A070678 A124091 A067849 * A294390 A152433 A094344

Adjacent sequences: A164265 A164266 A164267 * A164269 A164270 A164271

Keyword

sign

Author

Michael Somos, Aug 11 2009

Status

approved