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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.
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%I #21 Jul 31 2023 02:51:53

%S 2,16,32,45,60,79,91,105,120,136,154,165,180,197,211,225,240,256,272,

%T 285,300,318,331,345,360,376,393,405,420,437,451,465,480,496,512,525,

%U 540,564,571,585,600,616,635,645,660,677,691,705,720,736,752,765,780,798,811,825,840,856

%N 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.

%C Jan Vonk observed that a(n) ~ 15*n. - _Robin Visser_, Jul 30 2023

%H Robin Visser, <a href="/A344016/b344016.txt">Table of n, a(n) for n = 1..250</a>

%H Jan Vonk, <a href="https://doi.org/10.1090/bull/1700">Overconvergent modular forms and their explicit arithmetic</a>, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.

%e From _Robin Visser_, Jul 30 2023: (Start)

%e 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 - ....

%e The first coefficient factors as 356652 = 2^2 * 3^2 * 9907, so a(1) = 2.

%e The second coefficient factors as -16114360320000 = -2^16 * 3^5 * 5^4 * 1619, so a(2) = 16.

%e The third coefficient factors as 1298216343568384000000/3 = 2^32 * 3^-1 * 5^6 * 19344931, so a(3) = 32. (End)

%o (Sage)

%o def a(n):

%o j1 = sum([1]+[240*sigma(k,3)*x^k for k in range(1, 3*n)])

%o j2 = product([x]+[(1-x^k)^24 for k in range(1, 3*n)])

%o jinv = (j2/j1^3).taylor(x, 0, 3*n)

%o T3jinv = sum([jinv.coefficient(x^(3*k))*x^k for k in range(n+1)]+

%o [jinv.coefficient(x^k)*x^(3*k)/3 for k in range(n)])

%o for k in range(1, n):

%o c = T3jinv.taylor(x, 0, k).coefficient(x^k)

%o T3jinv -= c*(jinv^k)

%o coeff = T3jinv.taylor(x, 0, n).coefficient(x^n)

%o return Rational(coeff).valuation(2) # _Robin Visser_, Jul 30 2023

%Y Cf. A000521, A344014, A344015, A344017.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jun 17 2021

%E More terms from _Robin Visser_, Jul 30 2023