OFFSET
1,1
COMMENTS
It is not clear if all the coefficients are integers.
LINKS
Robin Visser, Table of n, a(n) for n = 1..200
Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
EXAMPLE
From Robin Visser, Jul 29 2023: (Start)
A q-expansion for the inverse of Klein's j-invariant is given by: j^-1 = q - 744*q^2 + 356652*q^3 - 140361152*q^4 + 49336682190*q^5 - 16114625669088*q^6 + O(q^7). Thus a q-expansion for U_2 operated on j^-1 is: U_2 j^-1 = -744*q - 140361152*q^2 - 16114625669088*q^3 + O(q^4).
Computing q-expansions for j^-2 and j^-3 gives j^-2 = q^2 - 1488*q^3 + O(q^4), and j^-3 = q^3 + O(q^4).
This yields an expansion for U_2 j^_1 in terms of powers of j^-1 as U_2 j^-1 = -744*j^-1 - 140914688*j^-2 - 16324041375744*j^-3 - ..., which gives the first three terms as a(1) = 744, a(2) = 140914688, and a(3) = 16324041375744. (End)
PROG
(Sage)
def a(n):
j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 2*n)])
j2 = product([x]+[(1-x^k)^24 for k in range(1, 2*n)])
jinv = (j2/j1^3).taylor(x, 0, 2*n)
U2jinv = sum([jinv.coefficient(x^(2*k))*x^k for k in range(0, 2*n)])
for k in range(1, n):
c = U2jinv.taylor(x, 0, k).coefficient(x^k)
U2jinv -= c*(jinv^k)
return -U2jinv.taylor(x, 0, n).coefficient(x^n) # Robin Visser, Jul 29 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 17 2021
EXTENSIONS
More terms from Robin Visser, Jul 29 2023
STATUS
approved