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
KEYWORD
sign
AUTHOR
Michael Somos, Sep 02 2014
STATUS
approved