A341304 - OEIS

%I #16 Jul 23 2023 17:07:20

%S 1,-84,-82,-456,4869,-2524,-10778,6888,-11150,4124,38304,81704,-71401,

%T -225288,99798,-40480,212016,37392,-419442,905352,141402,-690428,

%U -399258,-682032,-615607,936600,1813118,206968,-346416,-966028,1887670,-2220264,883796,2965868

%N Fourier coefficients of a modular form studied by Koike.

%C This is the form (1/t_{4a}) * ( 1-16*i/t_{4a} )*F_{4a}^8.  Here, F_{4a} is the hypergeometric function F(1/4, 1/2; 1; 32*i/t_{4a}).

%H Masao Koike, <a href="https://oeis.org/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, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.] See page 29.

%o (Sage)

%o def a(n):

%o     eta = x^(1/24)*product([(1 - x^k) for k in range(1, 2*n+1)])

%o     t4a = ((eta/eta(x=x^2))^12 - 64*(eta(x=x^2)/eta)^12) + 16*sqrt(-1)

%o     F4a = sum([rising_factorial(1/4,k)*rising_factorial(1/2,k)/

%o         (rising_factorial(1,k)^2)*((32*sqrt(-1))/t4a)^k for k in range(2*n+1)])

%o     f = (1/t4a)*(1 - 16*sqrt(-1)/t4a)*(F4a^8)

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

%Y Cf. A341305, A341306.

%K sign

%O 0,2

%A _N. J. A. Sloane_, Feb 13 2021

%E More terms from _Robin Visser_, Jul 23 2023