A091406 Reversion of series for j-function.

1, 744, 750420, 872769632, 1102652742882, 1470561136292880, 2037518752496883080, 2904264865530359889600, 4231393254051181981976079, 6273346050902229242859370584, 9433668720359866477436486024652

Offset: 1

Author

N. J. A. Sloane, Mar 03 2004

Keywords

• nonn
• easy

Comments

From Peter Bala, Dec 17 2013: (Start)

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).

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)

Invert j = 1/q + 744 + 196884*q + 21493760*q^2 + ... to get q = 1/j + 744/j^2 + 750420/j^3 + ....

Examples

			G.f. = x + 744*x^2 + 750420*x^3 + 872769632*x^4 + 1102652742882*x^5 + ... - _Michael Somos_, Dec 17 2021
		

References

• J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..300
• Youssef Abdelaziz and Jean-Marie Maillard, Modular forms, Schwarzian conditions, and symmetries of differential equations in physics, arXiv preprint arXiv:1611.08493, 2016
• Yang-Hui He and Vishnu Jejjala, Modular Matrix Models, arXiv:hep-th/0307293, 2003.
• Jean-Marie Maillard, Modular correspondences and replicable functions (unabridged version), arXiv:2505.16873 [math-ph], 2025. See pp. 6, 22.

Crossrefs

Cf. A000521, A178451. See A066396 for another version.

Programs

Maple

#A091406
with(numtheory):
Order := 12:
g2 := 4/3*(1 + 240*add(sigma[3](n)*q^n, n = 1..Order)):
g3 := 8/27*(1 - 504*add(sigma[5](n)*q^n, n = 1..Order)):
delta := series(g2^3 - 27*g3^2, q, Order):
#define the reciprocal of Klein's j_invariant
j_reciprocal := series(delta/(1728*g2^3), q, Order):
#find series reversion of j_reciprocal
j_inv := solve(series(j_reciprocal, q) = y, q):
seq(coeff(j_inv,y,n), n = 1..11); # Peter Bala, Dec 17 2013

Mathematica

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 *)
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 *)

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 */

Formula

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