%I #14 Jul 29 2023 03:28:38

%S 1,-3,-810,-14848,-123111,-544644,-1362010,-2330370,-3239838,2585060,

%T 15441024,4575528,56803975,51302133,-76460274,-7843816,58087692,

%U -680814720,178532126,-82694130,-573777270,317066108,1870315110,-1107862272,2815890921,240200154,1340122806

%N Fourier coefficients of the modular form (1/t_{6a}) * (1-6*sqrt(-3)/t_{6a}) * (1-12*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 23 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, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 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)*(1-6*sqrt(-3)/t6a)*(1-12*sqrt(-3)/t6a)*F6a^12

%o return f.taylor(x,0,n+1).coefficients()[n][0] # _Robin Visser_, Jul 23 2023

%K sign

%O 0,2

%A _Robert C. Lyons_, Feb 15 2021

%E More terms from _Robin Visser_, Jul 23 2023