A341566 Fourier coefficients of the modular form (1/t_{6a}^3) * (1-12*sqrt(-3) / t_{6a})^(1/6) * F_{6a}^10.

0, 1, -3, -27, 147, -183, -54, -423, 1434, 1323, -4141, -2580, -972, 20394, -1929, -20169, -15597, -7746, 58212, -27964, 76398, -19683, -117585, -94050, -60696, 474840, -103989, -89910, -72487, 148359, 100440, -1107363, 43554, 629289, 1257516, 517584, -745848, -1430837, -748068

Offset

0,3

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..38.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 0
    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*(phi12^3*(phi02^2 - phi12^2)^3)/8
    return f.taylor(x, 0, n+1).coefficient(x^(n+1/2))  # Robin Visser, Jul 31 2023

Crossrefs

Sequence in context: A285008 A001796 A174613 * A354655 A127509 A108142

Adjacent sequences: A341563 A341564 A341565 * A341567 A341568 A341569

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved