OFFSET
-1,2
COMMENTS
Number 9 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 22 2014
A generator (Hauptmodul) of the function field associated with the congruence subgroup Gamma_1(7). [Yang 2004] - Michael Somos, Jul 22 2014
REFERENCES
N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103). See p. 89 eq. (4.23)
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1003 from G. A. Edgar)
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 156.
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
Euler transform of period 7 sequence [ 3, -2, -1, -1, -2, 3, 0, ...]. - Michael Somos, Jul 22 2014
G.f.: (1/x) * Product_{k>0} (1 - x^(7*k - 2))^2 * (1 - x^(7*k - 5))^2 * (1 - x^(7*k - 3)) * (1 - x^(7*k - 4)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6)))^3.
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = v - w + u^2 - 2*w*v - 3*u*w + 4*u*v + w^2*v + u*w^2 - u^2*v - u^2*w^2 + 4*u^2*w - 4*u^2*w*v - 5*u*v^2 + 5*u*w*v^2.
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = 2*v * (u - 1) * (3*u*v + v - 2*u - 1) - (u^2 - v) * (u*v^2 - 2*u*v + 2*v + u - 1). - Michael Somos, Jul 22 2014
a(n) = A246713(n) unless n = 0. - Michael Somos, Sep 02 2014
G.f.: T(q) = 1/q + 3 + 4*q + ... for this sequence is cubically related to T7B(q) of A052240: T7B = T - 3 - 1/(T-1) - 1/T. - G. A. Edgar, Apr 12 2017 [corrected by Seiichi Manyama, Oct 10 2018]
EXAMPLE
G.f. = 1/q + 3 + 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)^{-3, 2, 1, 1, 2, -3, 0}[[Mod[k, 7, 1]]], {k, m}], {q, 0, n}]]]; (* Michael Somos, Jul 22 2014 *)
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2, q^7] QPochhammer[ q^5, q^7])^2 QPochhammer[ q^3, q^7] QPochhammer[ q^4, q^7] / (QPochhammer[ q^1, q^7] QPochhammer[ q^6, q^7])^3, {q, 0, n}]; (* Michael Somos, Jul 22 2014 *)
PROG
(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, -3, 2, 1, 1, 2, -3][k%7 + 1]), n))};
(Magma) A := Basis( ModularForms( Gamma1(7), 1), 58); B<q> := A[2] / A[3]; B; /* Michael Somos, Nov 09 2014 */
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 04 2005
STATUS
approved