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.

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.

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

Cf. A030181, A052240, A246713, A262933 (T/(T-1)), A305443 (1/T).

Sequence in context: A318830 A242803 A064460 * A367076 A078070 A254745

Adjacent sequences: A108478 A108479 A108480 * A108482 A108483 A108484

Keyword

sign

Author

Michael Somos, Jun 04 2005

Status

approved