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%I #21 Jul 30 2023 02:44:23
%S 744,140914688,16324041375744,1528926232501026816,
%T 127072326069001429975040,9781118992341002031206498304,
%U 713663408582010002941475567960064,50057559997415568004332170039751868416,3406371342315881760006472823773108302249984
%N Coefficients (negated) of expansion of the operator U_2 applied to j^-1, the inverse of the Klein j-invariant, with respect to powers of j^-1.
%C It is not clear if all the coefficients are integers.
%H Robin Visser, <a href="/A344014/b344014.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 29 2023: (Start)
%e 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).
%e 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).
%e 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)
%o (Sage)
%o def a(n):
%o j1 = sum([1]+[240*sigma(k,3)*x^k for k in range(1, 2*n)])
%o j2 = product([x]+[(1-x^k)^24 for k in range(1, 2*n)])
%o jinv = (j2/j1^3).taylor(x, 0, 2*n)
%o U2jinv = sum([jinv.coefficient(x^(2*k))*x^k for k in range(0, 2*n)])
%o for k in range(1, n):
%o c = U2jinv.taylor(x, 0, k).coefficient(x^k)
%o U2jinv -= c*(jinv^k)
%o return -U2jinv.taylor(x, 0, n).coefficient(x^n) # _Robin Visser_, Jul 29 2023
%Y Cf. A000521, A066395, A344015, A344016, A344017.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jun 17 2021
%E More terms from _Robin Visser_, Jul 29 2023
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