A341574 Fourier coefficients of the modular form (1/t_{6a}^3) * (1-6*sqrt(-3)/t_{6a}) * (1-12*sqrt(-3)/t_{6a})^(4/3) * F_{6a}^20.

0, 1, 3, -855, -19719, -189663, -809226, -255339, 9868542, 24108075, -19076611, -83250516, -178059060, -774131094, 1654990113, 4979928843, -8282963151, 7132715646, -6849297108, -29601661516, -8702922246, 87845108229, 5032903977, 141218051814, -264089426616, 302320735992, -825532125819

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..26.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 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)])
    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^3*(phix^2-phiy^2)^3*phix*(phix^2-9*phiy^2)*(phix^2+3*phiy^2)^4)/8
    return f.taylor(x, 0, n+1).coefficient(x^(n+1/2))  # Robin Visser, Jul 31 2023

Crossrefs

Sequence in context: A000723 A020525 A252762 * A283017 A093189 A163430

Adjacent sequences: A341571 A341572 A341573 * A341575 A341576 A341577

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved