A341567 Fourier coefficients of the modular form (1/t_{6a}) * (1-6*sqrt(-3)/t_{6a}) * (1-12*sqrt(-3)/t_{6a}) * F_{6a}^12.

1, -3, -810, -14848, -123111, -544644, -1362010, -2330370, -3239838, 2585060, 15441024, 4575528, 56803975, 51302133, -76460274, -7843816, 58087692, -680814720, 178532126, -82694130, -573777270, 317066108, 1870315110, -1107862272, 2815890921, 240200154, 1340122806

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 23 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Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 30.

Prog

(Sage)
def a(n):
    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)*(1-6*sqrt(-3)/t6a)*(1-12*sqrt(-3)/t6a)*F6a^12
    return f.taylor(x, 0, n+1).coefficients()[n][0]  # Robin Visser, Jul 23 2023

Crossrefs

Sequence in context: A259371 A294794 A293252 * A287695 A083250 A096086

Adjacent sequences: A341564 A341565 A341566 * A341568 A341569 A341570

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 23 2023

Status

approved