A341304 Fourier coefficients of a modular form studied by Koike.

1, -84, -82, -456, 4869, -2524, -10778, 6888, -11150, 4124, 38304, 81704, -71401, -225288, 99798, -40480, 212016, 37392, -419442, 905352, 141402, -690428, -399258, -682032, -615607, 936600, 1813118, 206968, -346416, -966028, 1887670, -2220264, 883796, 2965868

Offset: 0

Author

N. J. A. Sloane, Feb 13 2021

Keywords

• sign

Comments

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}).

Links

• Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.

Crossrefs

Cf. A341305, A341306.

Programs

Sage

def a(n):
    eta = x^(1/24)*product([(1 - x^k) for k in range(1, 2*n+1)])
    t4a = ((eta/eta(x=x^2))^12 - 64*(eta(x=x^2)/eta)^12) + 16*sqrt(-1)
    F4a = sum([rising_factorial(1/4,k)*rising_factorial(1/2,k)/
        (rising_factorial(1,k)^2)*((32*sqrt(-1))/t4a)^k for k in range(2*n+1)])
    f = (1/t4a)*(1 - 16*sqrt(-1)/t4a)*(F4a^8)
    return f.taylor(x,0,n+1).coefficients()[n][0]  # Robin Visser, Jul 23 2023

Extensions

More terms from Robin Visser, Jul 23 2023