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Expansion of a modular function for Gamma(7).
1

%I #9 Aug 03 2024 07:12:12

%S 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,

%T -1,12,18,10,-7,-21,-21,-3,20,30,17,-13,-37,-35,-4,36,53,30,-20,-62,

%U -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

%N Expansion of a modular function for Gamma(7).

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

%F 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.

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

%F 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

%o (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))}

%Y Cf. A375106, A375107.

%K sign

%O 0,9

%A _Michael Somos_, Jun 04 2005