A170991 Number of genus 2, degree n, simply ramified covers of an elliptic curve.

2, 16, 60, 160, 360, 672, 1240, 1920, 3180, 4400, 6832, 8736, 12880, 15840, 22320, 26112, 36666, 41040, 55720, 62720, 82104, 89056, 119520, 124800, 161980, 174240, 219744, 227360, 295920, 297600, 377952, 392832, 480420, 486080, 623820, 607392, 753160, 771680, 934800, 918400, 1157184

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..42.

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

Formula

G.f.: (5*E_2^3 - 3*E_2*E_4 - 2*E_6)/25920, 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 = (5*E2^3 - 3*E2*E4 - 2*E6)/25920
    return f.taylor(x, 0, n).coefficient(x^n)  # Robin Visser, Aug 08 2023

Crossrefs

Cf. A170992, A170993, A170994, A170995, A170996, A170997, A170998, A170999.

Sequence in context: A293620 A206980 A357816 * A209219 A207688 A208495

Adjacent sequences: A170988 A170989 A170990 * A170992 A170993 A170994

Keyword

nonn

Author

N. J. A. Sloane, Aug 31 2010

Extensions

More terms from Robin Visser, Aug 08 2023

Status

approved