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A170992
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Number of genus 3, degree n, simply ramified covers of an elliptic curve.
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8
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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
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OFFSET
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2,1
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COMMENTS
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The reference gives a generating function and the terms up to degree 18.
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LINKS
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FORMULA
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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