A344016 2-adic valuation of coefficients of expansion of the operator T_3 applied to j^-1, the inverse of the Klein j-invariant, with respect to powers of j^-1.

2, 16, 32, 45, 60, 79, 91, 105, 120, 136, 154, 165, 180, 197, 211, 225, 240, 256, 272, 285, 300, 318, 331, 345, 360, 376, 393, 405, 420, 437, 451, 465, 480, 496, 512, 525, 540, 564, 571, 585, 600, 616, 635, 645, 660, 677, 691, 705, 720, 736, 752, 765, 780, 798, 811, 825, 840, 856

Offset

1,1

Comments

Jan Vonk observed that a(n) ~ 15*n. - Robin Visser, Jul 30 2023

Links

Robin Visser, Table of n, a(n) for n = 1..250

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 30 2023: (Start)
An expansion for T_3 j^_1 in terms of powers of j^-1 is given by T_3 j^-1 = 356652*j^-1 - 16114360320000*j^-2 + 1298216343568384000000/3*j^-3 - ....
The first coefficient factors as 356652 = 2^2 * 3^2 * 9907, so a(1) = 2.
The second coefficient factors as -16114360320000 = -2^16 * 3^5 * 5^4 * 1619, so a(2) = 16.
The third coefficient factors as 1298216343568384000000/3 = 2^32 * 3^-1 * 5^6 * 19344931, so a(3) = 32. (End)

Prog

(Sage)
def a(n):
    j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 3*n)])
    j2 = product([x]+[(1-x^k)^24 for k in range(1, 3*n)])
    jinv = (j2/j1^3).taylor(x, 0, 3*n)
    T3jinv = sum([jinv.coefficient(x^(3*k))*x^k for k in range(n+1)]+
                 [jinv.coefficient(x^k)*x^(3*k)/3 for k in range(n)])
    for k in range(1, n):
        c = T3jinv.taylor(x, 0, k).coefficient(x^k)
        T3jinv -= c*(jinv^k)
    coeff = T3jinv.taylor(x, 0, n).coefficient(x^n)
    return Rational(coeff).valuation(2)  # Robin Visser, Jul 30 2023

Crossrefs

Cf. A000521, A344014, A344015, A344017.

Sequence in context: A109210 A056707 A069256 * A279034 A120069 A018975

Adjacent sequences: A344013 A344014 A344015 * A344017 A344018 A344019

Keyword

nonn

Author

N. J. A. Sloane, Jun 17 2021

Extensions

More terms from Robin Visser, Jul 30 2023

Status

approved