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%I #12 Aug 08 2023 09:13:38
%S 2,1456,91920,1931200,21639720,160786272,893985280,4001984640,
%T 15166797900,50211875600,149342289472,404551482816,1017967450960,
%U 2389725895200,5320611901440,11218821981312,22749778149786,44125038791280,83117833890400,150796252355840,267599811354744
%N Number of genus 4, 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.: (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
%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 = (355*E2^9 - 1395*E2^7*E4 - 600*E2^6*E6 + 1737*E2^5*E4^2 + 4410*E2^4*E4*E6
%o - 2145*E2^3*E4^3 - 1860*E2^3*E6^2 - 6300*E2^2*E4^2*E6 + 3600*E2*E4^4
%o + 4860*E2*E4*E6^2 - 2238*E4^3*E6 - 424*E6^3)/(2^15*3^9)
%o return f.taylor(x, 0, n).coefficient(x^n) # _Robin Visser_, Aug 08 2023
%Y Cf. A170991, A170992, 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