A230536 Expansion of q^(-1) * f(-q^5, -q^7) / f(-q, -q^11) in powers of q where f(,) is Ramanujan's two-variable theta function.

1, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, 0, 1, 2, 2, 2, 1, 0, -2, -3, -4, -4, -2, 0, 3, 5, 7, 6, 4, 0, -4, -8, -10, -9, -6, 0, 6, 12, 14, 14, 8, 0, -10, -18, -22, -20, -12, 0, 15, 26, 33, 29, 19, 0, -20, -37, -45, -42, -26, 0, 27, 52, 62, 58, 34, 0, -40, -72, -88

Offset

-1,14

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Number 13 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014

A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(12). [Yang 2004] - Michael Somos, Aug 07 2014

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

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 (c(q) / c(q^4) + phi(q) * psi(q^3) / (q * psi(q^6)^2)) / 2 = 2 / (c(q) / c(q^4) - phi(q) * psi(q^3) / (q * psi(q^6)^2)) in powers of q where c() is a cubic AGM theta function and phi(), psi() are Ramanujan theta functions.

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (1 + u^2).

G.f.: x^(-1) * (Product_{k>0} (1 - x^(12*k - 5)) * (1 - x^(12*k - 7)) / ((1 - x^(12*k - 1)) * (1 - x^(12*k - 11)))).

a(2*n) = A139139(n). a(6*n + 4) = 0. a(6*n + 2) = A139135(n).

Convolution inverse of A113306.

Example

G.f. = 1/q + 1 + q + q^2 + q^3 - q^6 - q^7 - q^8 - q^9 + q^11 + 2*q^12 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[ q^5, q^12] QPochhammer[ q^7, q^12] / (QPochhammer[ q, q^12] QPochhammer[ q^11, q^12]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/q Product[(1 - q^k)^-KroneckerSymbol[12, k], {k, n + 1}], {q, 0, n}]; (* Michael Somos, Aug 07 2014 *)

Prog

(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod( k=1, n, (1 - x^k)^-kronecker(12, k), 1 + x * O(x^n)), n))};

Crossrefs

Cf. A113306, A139135, A139139.

Sequence in context: A112215 A176389 A076451 * A306257 A357316 A108839

Adjacent sequences: A230533 A230534 A230535 * A230537 A230538 A230539

Keyword

sign

Author

Michael Somos, Oct 22 2013

Status

approved