A245421 Expansion of q^(-1) * f(-q^2, -q^7) * f(-q^4, -q^5) / f(-q, -q^8)^2 in powers of q where f() is Ramanujan's two-variable theta function.

1, 2, 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

-1,2

Comments

Number 11 of the 15 generalized eta-quotients listed in Table I of Yang 2004.

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(9). [Yang 2004]

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. A. Edgar, Table of n, a(n) for n = -1..1003

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.

Formula

Expansion of q^(-1) * f(-q) * f(-q^9)^3 / (f(-q^3) * f(-q, -q^8)^3) in powers of q where f() is a Ramanujan theta function.

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

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

a(n) = A245424(n) unless n=0.

G.f. T(q) = 1/q + 2 + 2*q + ... for this function is cubically related to T9B(q) of A058091: T9B = T - 2 - 1/T - 1/(T-1). - G. A. Edgar, Apr 13 2017

Example

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

Crossrefs

Cf. A245424.

Sequence in context: A237593 A338169 A243847 * A134143 A295555 A085684

Adjacent sequences: A245418 A245419 A245420 * A245422 A245423 A245424

Keyword

sign

Author

Michael Somos, Jul 21 2014

Status

approved