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A108481
Expansion of q^(-1) * f(-q^2, -q^5)^2 * f(-q^3, -q^4) / f(-q^1, -q^6)^3 in powers of q where f() is Ramanujan's two-variable theta function.
5
1, 3, 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, 4992, 8968, 6251, -2506, -11285, -12579
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.
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
Cf. A030181, A052240, A246713, A262933 (T/(T-1)), A305443 (1/T).
Sequence in context: A318830 A242803 A064460 * A367076 A078070 A254745
KEYWORD
sign
AUTHOR
Michael Somos, Jun 04 2005
STATUS
approved