A108482 Expansion of a modular function for Gamma(7).

1, 1, 1, 0, -1, -1, 0, 1, 2, 1, -1, -3, -3, 0, 3, 5, 3, -2, -6, -6, -1, 6, 9, 5, -4, -12, -11, -1, 12, 18, 10, -7, -21, -21, -3, 20, 30, 17, -13, -37, -35, -4, 36, 53, 30, -20, -62, -59, -8, 57, 85, 47, -35, -101, -95, -11, 94, 138, 78, -54, -159, -150, -19, 145, 213, 118, -85, -247, -231, -27, 225, 330, 183, -128, -375

Offset: 0

Author

Michael Somos, Jun 04 2005

Keywords

• sign

References

• W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 157.

Crossrefs

Cf. A375106, A375107.

Programs

PARI

{a(n)=if(n<0, 0, polcoeff( prod(k=1,n,(1-x^k+x*O(x^n))^[0,-1,0,1,1,0,-1][k%7+1]), n))}

Formula

Given g.f. A(x), then B(x)=x^-3*A(x^7) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^7 -v^7 +u*v^3 +u^9*v^6 +u^2*v^6 +3*u^5*v^8 -u^3*v^2 -u^3*v^9 -u^4*v^5 -u^5*v -5*u^6*v^4 -3*u^7*v^7 -2*u^8*v^3.

G.f.: Product_{k>0} (1-x^(7k-3))(1-x^(7k-4))/((1-x^(7k-1))(1-x^(7k-6))).

G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375107. - Seiichi Manyama, Aug 03 2024