A245424 Expansion of q^(-1) * f(-q^4, -q^5)^2 / (f(-q, -q^8) * f(-q^2, -q^7)) in powers of q where f() is Ramanujan's two-variable theta function.

1, 1, 2, 2, 1, -1, -2, -3, -2, 1, 4, 6, 5, 1, -5, -11, -12, -7, 3, 15, 22, 19, 5, -15, -32, -36, -22, 8, 40, 58, 50, 12, -41, -84, -93, -54, 22, 103, 148, 124, 32, -96, -200, -219, -128, 46, 231, 330, 275, 67, -216, -439, -477, -275, 107, 501, 708, 590, 146

Offset

-1,3

Comments

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(9).

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 q^(-1) * f(-q^4, -q^5)^3 * f(-q^3) / (f(-q) * f(-q^9)^3) in powers of q where f() is a Ramanujan theta function.

Euler transform of period 9 sequence [ 1, 1, 0, -2, -2, 0, 1, 1, 0, ...].

Given g.f. A(q), then 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 - v) * (u*v^2 + v^2 + 1) - 2 * (u + v + u*v)^2.

Given g.f. A(q), then 0 = f(A(q), A(q^3)) where f(u, v) = (v^2 + v + 1) * (u^3 - v) - 3*u*v * (u + 1) * (v + 2).

a(n) = A245421(n) unless n=0.

Example

G.f. = 1/q + 1 + 2*q + 2*q^2 + q^3 - q^4 - 2*q^5 - 3*q^6 - 2*q^7 + q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9])^3 QPochhammer[ q^3] / (QPochhammer[ q] ), {q, 0, n}];
a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-1, -1, 0, 2, 2, 0, -1, -1, 0}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9])^2 / (QPochhammer[ q^1, q^9] QPochhammer[ q^2, q^9] QPochhammer[ q^7, q^9] QPochhammer[ q^8, q^9] ), {q, 0, n}];

Prog

(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, -1, 0, 2, 2, 0, -1, -1][k%9 + 1]), n))};

Crossrefs

Cf. A245421.

Sequence in context: A301565 A237265 A035463 * A071784 A161638 A066030

Adjacent sequences: A245421 A245422 A245423 * A245425 A245426 A245427

Keyword

sign

Author

Michael Somos, Jul 21 2014

Status

approved