A341563 Fourier coefficients of the modular form (1/t_{6a}) * sqrt( 1-12*sqrt(-3)/t_{6a} ) * F_{6a}^6.

1, 15, 54, -88, -423, 540, -418, 810, 594, 836, -1320, -4104, -209, 1431, -594, 4256, 8100, -4752, -298, -6270, 17226, -12100, -22842, -1296, -9063, 8910, 19494, 29160, 12540, -7668, -34738, 37224, -22572, 21812, -61560, -46872, 67562, -3135, -47520, -76912, 85617, 67716, 32076

Offset

0,2

Comments

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 31 2023

Links

Table of n, a(n) for n=0..42.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. Sloane wrote 2005 on the first page but the internal evidence suggests 1997.] See page 30.

Prog

(Sage)
def a(n):
    if n==0: return 1
    theta2 = sum([1]+[2*x^(k^2/2) for k in range(1, n+1)])
    theta3 = sum([2*x^((k^2 + k + 1/4)/2) for k in range(n)])
    phi0 = theta2(x=x^2)*theta2(x=x^6) + theta3(x=x^2)*theta3(x=x^6)
    phi1 = theta2(x=x^2)*theta3(x=x^6) + theta3(x=x^2)*theta2(x=x^6)
    phi02, phi12 = phi0(x=x^2), phi1(x=x^2)
    f = phi0*(phi12*(phi02^2 - phi12^2)*(phi02^2 + 3*phi12^2))/2
    return f.taylor(x, 0, n+1).coefficient(x^(n+1/2))  # Robin Visser, Jul 31 2023

Crossrefs

Sequence in context: A358398 A219384 A198955 * A063436 A010004 A172073

Adjacent sequences: A341560 A341561 A341562 * A341564 A341565 A341566

Keyword

sign

Author

Robert C. Lyons, Feb 14 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved