

A091406


Reversion of series for jfunction.


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:
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^2 + ... 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 + ... to get q = 1/j + 744/j^2 + 750420/j^2 + ...


REFERENCES

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


LINKS

Table of n, a(n) for n=1..11.
Y.H. He and V. Jejjala, Modular Matrix Models.


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 = InverseSeries[ Series[ s1, {q, 0, max} ], j] /. j > 1/x; Rest[ CoefficientList[ s2, x ] ](* JeanFrançois Alcover, Feb 16 2012 *)


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



