A341570 Fourier coefficients of the modular form (1/t_{6a}^3) * (1-12*sqrt(-3)/t_{6a})^(5/6) * F_{6a}^14.

0, 1, 21, 117, -261, -3399, 1098, 40905, -22710, -206181, 247739, -120612, 868644, -1185030, 506703, -516537, -3919509, 5256222, 1748340, 14399300, -17795346, -15633891, -3511593, -11977218, 31605912, 59669784, 78298419, -83869830, -214809535, -81823593, 192014280, -24742755

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..31.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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)])
    phi0 = theta2(x=x^2)*theta2(x=x^6) + theta3(x=x^2)*theta3(x=x^6)
    phi1 = theta2(x=x^2)*theta3(x=x^6) + theta3(x=x^2)*theta2(x=x^6)
    phi02, phi12 = phi0(x=x^2), phi1(x=x^2)
    f = phi0*(phi12^3*(phi02^2 - phi12^2)^3*(phi02^2 + 3*phi12^2)^2)/8
    return f.taylor(x, 0, n+1).coefficient(x^(n+1/2))  # Robin Visser, Jul 31 2023

Crossrefs

Sequence in context: A306532 A219154 A054257 * A297316 A079417 A267877

Adjacent sequences: A341567 A341568 A341569 * A341571 A341572 A341573

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved