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A091406 Reversion of series for j-function. 5
1, 744, 750420, 872769632, 1102652742882, 1470561136292880, 2037518752496883080, 2904264865530359889600, 4231393254051181981976079, 6273346050902229242859370584, 9433668720359866477436486024652 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 + ....

REFERENCES

Y Abdelaziz, JM Maillard, Modular forms, Schwarzian conditions, and symmetries of differential equations in physics, arXiv preprint arXiv:1611.08493, 2016

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

Y.-H. He and V. Jejjala, Modular Matrix Models, arXiv:hep-th/0307293, 2003.

FORMULA

a(n) ~ c * 1728^n / n^(3/2), where c = 0.000873226754634291459391356302898297243945046378336447143... - Vaclav Kotesovec, Jun 28 2017

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-Fran├žois Alcover, Feb 16 2012, fixed by Vaclav Kotesovec, Jun 28 2017 *)

PROG

(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 */

CROSSREFS

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

Sequence in context: A178451 A066395 A161557 * A066396 A099819 A051978

Adjacent sequences:  A091403 A091404 A091405 * A091407 A091408 A091409

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mar 03 2004

STATUS

approved

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Last modified November 18 16:40 EST 2017. Contains 294894 sequences.