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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.
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%I #22 Aug 01 2023 11:00:50

%S 1,18,33,47,61,80,92,107,121,138,155,167,181,199,212,227,241,258,273,

%T 287,301,322,332,347,361,378,394,407,421,439,452,467,481,498,513,527,

%U 541,560,572,587,601,618,637,647,661,679,692,707,721,738,753,767,781,801,812,827,841,858

%N 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.

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

%H Robin Visser, <a href="/A344017/b344017.txt">Table of n, a(n) for n = 1..200</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_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 - ....

%e The first coefficient factors as 49336682190 = 2 * 3 * 5 * 1644556073, so a(1) = 1.

%e The second coefficient factors as -122566701099729715200000 = -2^18 * 3^4 * 5^5 * 1847130309301, so a(2) = 18.

%e The third coefficient factors as 177278377115100363578123747328000000 = 2^33 * 3^7 * 5^6 * 31 * 47 * 414512421715739, so a(3) = 33. (End)

%o (Sage)

%o def a(n):

%o j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 5*n)])

%o j2 = product([x]+[(1-x^k)^24 for k in range(1, 5*n)])

%o jinv = (j2/j1^3).taylor(x, 0, 5*n)

%o T5jinv = sum([jinv.coefficient(x^(5*k))*x^k for k in range(n+1)]+

%o [jinv.coefficient(x^k)*x^(5*k)/5 for k in range(n)])

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

%o c = T5jinv.taylor(x, 0, k).coefficient(x^k)

%o T5jinv -= c*(jinv^k)

%o coeff = T5jinv.taylor(x, 0, n).coefficient(x^n)

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

%Y Cf. A000521, A344014, A344015, A344016.

%K nonn

%O 1,2

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

%E More terms from _Robin Visser_, Jul 30 2023