A341573 Fourier coefficients of the modular form (1/t_{6a}^3) * (1-12*sqrt(-3)/t_{6a})^(3/2) * F_{6a}^18.

0, 1, 45, 837, 7971, 36009, 6858, -622215, -1749510, 3437451, 17561891, -11031732, -67208940, 69554250, -49492953, -261597033, 1059514371, -637727490, 1136241540, -3548694460, 1012385646, -1118737251, -2102447745, 12785364702, 10428178248, -5328737064, -24564279141, -20702406870

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..27.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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

Crossrefs

Sequence in context: A024379 A134290 A359933 * A078761 A034967 A199352

Adjacent sequences: A341570 A341571 A341572 * A341574 A341575 A341576

Keyword

sign

Author

Robert C. Lyons, Feb 15 2021

Extensions

More terms from Robin Visser, Jul 31 2023

Status

approved