A341571 Fourier coefficients of the modular form (1/t_{6a}) * (1-6*sqrt(-3)/t_{6a}) * (1-12*sqrt(-3)/t_{6a})^(5/3) * F_{6a}^16.

1, 21, -666, -34048, -655335, -7427988, -55630666, -290463138, -1104081822, -3239870284, -7842842112, -15742299384, -24417702377, -30080972403, -28964466882, 22837159352, 121791163356, 164707345152, 491149796174, 668773471470, 222330211434, 722181385964, 715812714486

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..22.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 31.

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)^5)/2
    return f.taylor(x, 0, n+1).coefficient(x^(n+1/2))  # Robin Visser, Jul 31 2023

Crossrefs

Sequence in context: A297312 A158216 A262960 * A020246 A239099 A006934

Adjacent sequences: A341568 A341569 A341570 * A341572 A341573 A341574

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved