A170992 Number of genus 3, degree n, simply ramified covers of an elliptic curve.

2, 160, 2448, 18304, 90552, 341568, 1068928, 2877696, 7014204, 15423200, 32107456, 61663104, 115156144, 200764608, 346235904, 561158400, 911313450, 1395016992, 2158796512, 3161199104, 4703221224, 6631046848, 9570587136, 13069048320, 18346507756, 24453721152, 33515655552

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..28.

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

Formula

G.f.: (-6*E_2^6 + 15*E_2^4*E_4 + 4*E_2^3*E_6 - 12*E_2^2*E_4^2 - 12*E_2*E_4*E_6 + 7*E_4^3 + 4*E_6^2)/1492992, 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 = (-6*E2^6 + 15*E2^4*E4 + 4*E2^3*E6 - 12*E2^2*E4^2
        - 12*E2*E4*E6 + 7*E4^3 + 4*E6^2)/1492992
    return f.taylor(x, 0, n).coefficient(x^n)  # Robin Visser, Aug 08 2023

Crossrefs

Cf. A170991, A170993, A170994, A170995, A170996, A170997, A170998, A170999.

Sequence in context: A064071 A116638 A326488 * A378781 A179958 A272244

Adjacent sequences: A170989 A170990 A170991 * A170993 A170994 A170995

Keyword

nonn

Author

N. J. A. Sloane, Aug 31 2010

Extensions

More terms from Robin Visser, Aug 08 2023

Status

approved