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A091406 Reversion of series for j-function.

%I A091406 #48 May 30 2025 10:50:50
%S A091406 1,744,750420,872769632,1102652742882,1470561136292880,
%T A091406 2037518752496883080,2904264865530359889600,4231393254051181981976079,
%U A091406 6273346050902229242859370584,9433668720359866477436486024652
%N A091406 Reversion of series for j-function.
%C A091406 From _Peter Bala_, Dec 17 2013: (Start)
%C A091406 Given a formal Laurent series L(z) = 1/z + a + b*z + c*z^2 + ..., there exists a formal series L^<-1>(z) = 1/z + A/z^2 + B/z^3 + ... such that L(L^<-1>(z)) = L^<-1>(L(z)) = z. The series L^<-1>(z) is called the reversion of the series L(z).
%C A091406 To find L^<-1>(z), first find the series reversion of the reciprocal series 1/L(z) = z - a*z^2 + z^3*(a^2 - b) - ... with respect to z, and then replace the variable z with the variable 1/z. This is the approach used in the Maple program below. (End)
%C A091406 Invert j = 1/q + 744 + 196884*q + 21493760*q^2 + ... to get q = 1/j + 744/j^2 + 750420/j^3 + ....
%D A091406 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.
%H A091406 Vaclav Kotesovec, <a href="/A091406/b091406.txt">Table of n, a(n) for n = 1..300</a>
%H A091406 Youssef Abdelaziz and Jean-Marie Maillard, <a href="http://arxiv.org/abs/1611.08493">Modular forms, Schwarzian conditions, and symmetries of differential equations in physics</a>, arXiv preprint arXiv:1611.08493, 2016
%H A091406 Yang-Hui He and Vishnu Jejjala, <a href="http://arXiv.org/abs/hep-th/0307293">Modular Matrix Models</a>, arXiv:hep-th/0307293, 2003.
%H A091406 Jean-Marie Maillard, <a href="https://arxiv.org/abs/2505.16873">Modular correspondences and replicable functions (unabridged version)</a>, arXiv:2505.16873 [math-ph], 2025. See pp. 6, 22.
%F A091406 a(n) ~ c * 1728^n / n^(3/2), where c = 2 * Gamma(3/4)^4 / (exp(2*Pi) * sqrt(3) * Pi^(3/2)) = 0.000873226754634291459391356302898297243945046378336447143... - _Vaclav Kotesovec_, Jun 28 2017, updated Mar 07 2018
%e A091406 G.f. = x + 744*x^2 + 750420*x^3 + 872769632*x^4 + 1102652742882*x^5 + ... - _Michael Somos_, Dec 17 2021
%p A091406 #A091406
%p A091406 with(numtheory):
%p A091406 Order := 12:
%p A091406 g2 := 4/3*(1 + 240*add(sigma[3](n)*q^n, n = 1..Order)):
%p A091406 g3 := 8/27*(1 - 504*add(sigma[5](n)*q^n, n = 1..Order)):
%p A091406 delta := series(g2^3 - 27*g3^2, q, Order):
%p A091406 #define the reciprocal of Klein's j_invariant
%p A091406 j_reciprocal := series(delta/(1728*g2^3), q, Order):
%p A091406 #find series reversion of j_reciprocal
%p A091406 j_inv := solve(series(j_reciprocal, q) = y, q):
%p A091406 seq(coeff(j_inv,y,n), n = 1..11); # _Peter Bala_, Dec 17 2013
%t A091406 max = 9; s1 = 1728*Series[ KleinInvariantJ[t], {t, 0, 2*max} ] /. t -> -2*I*(Pi/Log[q]); s2 = Normal[InverseSeries[ Series[ s1, {q, 0, max} ], j]] /. j -> 1/x; Rest[ CoefficientList[ s2, x ] ] (* _Jean-François Alcover_, Feb 16 2012, fixed by _Vaclav Kotesovec_, Jun 28 2017 *)
%t A091406 a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ 1/(1728 KleinInvariantJ[ Log[q]/(2 Pi I)]), {q, 0, n}]], {q, 0, n}]]; Table[a[n], {n, 12}] (* _Michael Somos_, Dec 17 2021 *)
%o A091406 (PARI) {a(n) = local(A); if( n<1, 0, A = O(x^n); A = x * (eta(x^2 + A) / eta(x + A))^24; polcoeff( serreverse( A / (1 + 256*A)^3), n))} /* _Michael Somos_, Jul 13 2004 */
%Y A091406 Cf. A000521, A178451. See A066396 for another version.
%K A091406 nonn,easy
%O A091406 1,2
%A A091406 _N. J. A. Sloane_, Mar 03 2004