A170992 - OEIS

%I #13 Aug 08 2023 09:13:34

%S 2,160,2448,18304,90552,341568,1068928,2877696,7014204,15423200,

%T 32107456,61663104,115156144,200764608,346235904,561158400,911313450,

%U 1395016992,2158796512,3161199104,4703221224,6631046848,9570587136,13069048320,18346507756,24453721152,33515655552

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

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

%H Mike Roth and Noriko Yu, <a href="https://inspirehep.net/literature/1393371">Mirror Symmetry for Elliptic Curves: The A-Model (Fermionic) Counting</a>, Clay Mathematics Proceedings, Volume 11, 2010.

%F 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

%o (Sage)

%o def a(n):

%o     E2 = sum([1]+[-24*sigma(k)*x^k for k in range(1, n+1)])

%o     E4 = sum([1]+[240*sigma(k,3)*x^k for k in range(1, n+1)])

%o     E6 = sum([1]+[-504*sigma(k,5)*x^k for k in range(1, n+1)])

%o     f = (-6*E2^6 + 15*E2^4*E4 + 4*E2^3*E6 - 12*E2^2*E4^2

%o         - 12*E2*E4*E6 + 7*E4^3 + 4*E6^2)/1492992

%o     return f.taylor(x, 0, n).coefficient(x^n)  # _Robin Visser_, Aug 08 2023

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

%K nonn

%O 2,1

%A _N. J. A. Sloane_, Aug 31 2010

%E More terms from _Robin Visser_, Aug 08 2023