A341565 Fourier coefficients of the modular form (1/t_{6a}) * (1-12*sqrt(-3)/t_{6a})^(7/6) * F_{6a}^10.

1, 39, 630, 5336, 24201, 48636, -9010, -130950, -28494, -536860, -1191576, 2163096, -1089665, 1915839, 5242734, 1311824, -7589916, 7560720, -14913082, -3150750, -14651190, 8250716, -7614810, -8887536, 25910649, 73227294, -21473658, 59100840, 47646780, -125614836, -7751458

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..30.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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

Crossrefs

Sequence in context: A142976 A200409 A363836 * A034187 A059609 A010955

Adjacent sequences: A341562 A341563 A341564 * A341566 A341567 A341568

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved