%I #13 Jul 24 2023 12:08:36
%S 0,1,-45,297,-759,1089,-1674,405,4446,9099,-22867,-54900,24300,273546,
%T 34353,-619893,-574047,748350,2466828,-1316812,-3016134,-1623483,
%U 5335065,5699430,-10936728,-3359880,-1083339,24803010,2339387,-28526913,-6509160,-19884183,60838470,-22877559,6057828
%N Fourier coefficients of the modular form (1/t_{6a}^3) * (1-6*sqrt(-3)/t_{6a}) * F_{6a}^12.
%C Here, F_{6a} is the hypergeometric function F(1/3, 1/2; 1; 12*sqrt(-3)/t_{6a}). The definition given on page 23 in the linked manuscript has a minor typo where "t_{3A}" should be "t_{6a}". - _Robin Visser_, Jul 24 2023
%H Masao Koike, <a href="/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. Sloane wrote 2005 on the first page but the internal evidence suggests 1997.] See page 30.
%o (Sage)
%o def a(n):
%o if n==0: return 0
%o eta = x^(1/24)*product([(1 - x^k) for k in range(1, 2*n+1)])
%o t6a = ((eta(x=x^2)/eta(x=x^6))^6
%o - 27*(eta(x=x^6)/eta(x=x^2))^6)(x=sqrt(x)) + 6*sqrt(-3)
%o F6a = sum([rising_factorial(1/3, k)*rising_factorial(1/2, k)/
%o (rising_factorial(1, k)^2)*((12*sqrt(-3))/t6a)^k for k in range(2*n+1)])
%o f = (1/t6a^3)*(1-6*sqrt(-3)/t6a)*F6a^12
%o return f.taylor(x,0,n+1).coefficients()[n-1][0] # _Robin Visser_, Jul 24 2023
%K sign
%O 0,3
%A _Robert C. Lyons_, Feb 15 2021
%E More terms from _Robin Visser_, Jul 24 2023