A170993 Number of genus 4, degree n, simply ramified covers of an elliptic curve.

2, 1456, 91920, 1931200, 21639720, 160786272, 893985280, 4001984640, 15166797900, 50211875600, 149342289472, 404551482816, 1017967450960, 2389725895200, 5320611901440, 11218821981312, 22749778149786, 44125038791280, 83117833890400, 150796252355840, 267599811354744

Offset

2,1

Comments

The reference gives a generating function and the terms up to degree 18.

Links

Table of n, a(n) for n=2..22.

Mike Roth and Noriko Yu, Mirror Symmetry for Elliptic Curves: The A-Model (Fermionic) Counting, Clay Mathematics Proceedings, Volume 11, 2010.

Formula

G.f.: (355*E_2^9 - 1395*E_2^7*E_4 - 600*E_2^6*E_6 + 1737*E_2^5*E_4^2 + 4410*E_2^4*E_4*E_6 - 2145*E_2^3*E_4^3 - 1860*E_2^3*E_6^2 - 6300*E_2^2*E_4^2*E_6 + 3600*E_2*E_4^4 + 4860*E_2*E_4*E_6^2 - 2238*E_4^3*E_6 - 424*E_6^3)/(2^15 * 3^9), where E_k = 1 - (2*k/B_k)*Sum_{i > 0} Sum_{d dividing i} d^(k-1)*q^i is the Eisenstein series of weight k. - Robin Visser, Aug 08 2023

Prog

(Sage)
def a(n):
    E2 = sum([1]+[-24*sigma(k)*x^k for k in range(1, n+1)])
    E4 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, n+1)])
    E6 = sum([1]+[-504*sigma(k, 5)*x^k for k in range(1, n+1)])
    f = (355*E2^9 - 1395*E2^7*E4 - 600*E2^6*E6 + 1737*E2^5*E4^2 + 4410*E2^4*E4*E6
         - 2145*E2^3*E4^3 - 1860*E2^3*E6^2 - 6300*E2^2*E4^2*E6 + 3600*E2*E4^4
         + 4860*E2*E4*E6^2 - 2238*E4^3*E6 - 424*E6^3)/(2^15*3^9)
    return f.taylor(x, 0, n).coefficient(x^n)  # Robin Visser, Aug 08 2023

Crossrefs

Cf. A170991, A170992, A170994, A170995, A170996, A170997, A170998, A170999.

Sequence in context: A272168 A173131 A160087 * A259323 A172939 A172234

Adjacent sequences: A170990 A170991 A170992 * A170994 A170995 A170996

Keyword

nonn

Author

N. J. A. Sloane, Aug 31 2010

Extensions

More terms from Robin Visser, Aug 08 2023

Status

approved