A341564 Fourier coefficients of the modular form (1-6*sqrt(-3)/t_{6a}) * (1/t_{6a}) * (1-12*sqrt(-3)/t_{6a})^(1/3) * F_{6a}^8.

1, -27, -378, -832, 729, -2484, 14870, 10206, -22302, -16300, 22464, -115128, 64759, -19683, 157086, -16456, 67068, 314496, -149266, -401490, -241110, -443188, -275562, 922752, -131319, 602154, -697626, 938952, 440100, 870156, 2067062, -606528, -5620860, -1680748, 3108456

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..34.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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)])
    phix = theta2(x=x^4)*theta2(x=x^12) + theta3(x=x^4)*theta3(x=x^12)
    phiy = theta2(x=x^4)*theta3(x=x^12) + theta3(x=x^4)*theta2(x=x^12)
    f = (phiy*(phix^2 - phiy^2)*phix*(phix^2 - 9*phiy^2)*(phix^2 + 3*phiy^2))/2
    return f.taylor(x, 0, n+1).coefficient(x^(n+1/2))  # Robin Visser, Jul 31 2023

Crossrefs

Sequence in context: A161954 A162369 A231858 * A162726 A010979 A022591

Adjacent sequences: A341561 A341562 A341563 * A341565 A341566 A341567

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved