A246713 Expansion of q^(-1) * f(-q^3, -q^4)^3 / (f(-q^1, -q^6)^2 * f(-q^2, -q^5)) in powers of q where f() is Ramanujan's two-variable theta function.

1, 2, 4, 3, 0, -5, -7, -2, 8, 16, 12, -7, -29, -35, -10, 37, 70, 53, -21, -106, -126, -38, 119, 226, 164, -70, -326, -378, -106, 353, 652, 469, -189, -885, -1015, -290, 910, 1664, 1179, -483, -2205, -2492, -692, 2212, 3998, 2809, -1120, -5119, -5754, -1598

Offset

-1,2

Comments

A generator (Hauptmodul) of the function field associated with the congruence subgroup Gamma_1(7).

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

Euler transform of period 7 sequence [ 2, 1, -3, -3, 1, 2, 0, ...].

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 + v) * (u*v^2 + v^2 - 1) - 2*v * (u + 1) * (v^2 + 2*u*v + v + 3*u).

a(n) = A108481(n) unless n=0.

Example

G.f. = 1/q + 2 + 4*q + 3*q^2 - 5*q^4 - 7*q^5 - 2*q^6 + 8*q^7 + 16*q^8 + ...

Mathematica

a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-2, -1, 3, 3, -1, -2, 0}[[Mod[k, 7, 1]]], {k, m}], {q, 0, n}]]];
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3, q^7] QPochhammer[ q^4, q^7])^3 / (QPochhammer[ q^1, q^7]^2 QPochhammer[ q^2, q^7] QPochhammer[ q^5, q^7] QPochhammer[ q^6, q^7]^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, 3, 3, -1, -2][k%7 + 1]), n))};

Crossrefs

Cf. A108481.

Sequence in context: A351427 A201911 A048644 * A106137 A213381 A127651

Adjacent sequences: A246710 A246711 A246712 * A246714 A246715 A246716

Keyword

sign

Author

Michael Somos, Sep 02 2014

Status

approved