A344017 2-adic valuation of coefficients of expansion of the operator T_5 applied to j^-1, the inverse of the Klein j-invariant, with respect to powers of j^-1.

1, 18, 33, 47, 61, 80, 92, 107, 121, 138, 155, 167, 181, 199, 212, 227, 241, 258, 273, 287, 301, 322, 332, 347, 361, 378, 394, 407, 421, 439, 452, 467, 481, 498, 513, 527, 541, 560, 572, 587, 601, 618, 637, 647, 661, 679, 692, 707, 721, 738, 753, 767, 781, 801, 812, 827, 841, 858

Offset

1,2

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..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 30 2023: (Start)
An expansion for T_5 j^_1 in terms of powers of j^-1 is given by T_5 j^-1 = 49336682190*j^-1 - 122566701099729715200000*j^-2 + 177278377115100363578123747328000000*j^-3 - ....
The first coefficient factors as 49336682190 = 2 * 3 * 5 * 1644556073, so a(1) = 1.
The second coefficient factors as -122566701099729715200000 = -2^18 * 3^4 * 5^5 * 1847130309301, so a(2) = 18.
The third coefficient factors as 177278377115100363578123747328000000 = 2^33 * 3^7 * 5^6 * 31 * 47 * 414512421715739, so a(3) = 33. (End)

Prog

(Sage)
def a(n):
    j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 5*n)])
    j2 = product([x]+[(1-x^k)^24 for k in range(1, 5*n)])
    jinv = (j2/j1^3).taylor(x, 0, 5*n)
    T5jinv = sum([jinv.coefficient(x^(5*k))*x^k for k in range(n+1)]+
                 [jinv.coefficient(x^k)*x^(5*k)/5 for k in range(n)])
    for k in range(1, n):
        c = T5jinv.taylor(x, 0, k).coefficient(x^k)
        T5jinv -= c*(jinv^k)
    coeff = T5jinv.taylor(x, 0, n).coefficient(x^n)
    return Rational(coeff).valuation(2)  # Robin Visser, Jul 30 2023

Crossrefs

Cf. A000521, A344014, A344015, A344016.

Sequence in context: A299707 A350958 A045556 * A049731 A188212 A093478

Adjacent sequences: A344014 A344015 A344016 * A344018 A344019 A344020

Keyword

nonn

Author

N. J. A. Sloane, Jun 17 2021

Extensions

More terms from Robin Visser, Jul 30 2023

Status

approved