A341568 Fourier coefficients of the modular form (1/t_{6a}^3) * (1-6*sqrt(-3)/t_{6a}) * F_{6a}^12.

0, 1, -45, 297, -759, 1089, -1674, 405, 4446, 9099, -22867, -54900, 24300, 273546, 34353, -619893, -574047, 748350, 2466828, -1316812, -3016134, -1623483, 5335065, 5699430, -10936728, -3359880, -1083339, 24803010, 2339387, -28526913, -6509160, -19884183, 60838470, -22877559, 6057828

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 24 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Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 0
    eta = x^(1/24)*product([(1 - x^k) for k in range(1, 2*n+1)])
    t6a = ((eta(x=x^2)/eta(x=x^6))^6
      - 27*(eta(x=x^6)/eta(x=x^2))^6)(x=sqrt(x)) + 6*sqrt(-3)
    F6a = sum([rising_factorial(1/3, k)*rising_factorial(1/2, k)/
      (rising_factorial(1, k)^2)*((12*sqrt(-3))/t6a)^k for k in range(2*n+1)])
    f = (1/t6a^3)*(1-6*sqrt(-3)/t6a)*F6a^12
    return f.taylor(x, 0, n+1).coefficients()[n-1][0]  # Robin Visser, Jul 24 2023

Crossrefs

Sequence in context: A296326 A064561 A291177 * A093761 A324458 A073873

Adjacent sequences: A341565 A341566 A341567 * A341569 A341570 A341571

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 24 2023

Status

approved