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A263189
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Coefficient of y^0 in G(x,y)^4 where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n.
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1
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1, -12, 18, 20, 24, -96, -4, -120, 138, -116, 456, -132, 356, -540, 900, -1884, 1440, -2076, 1546, -3204, 5772, -6572, 7860, -7440, 12240, -13128, 15828, -18632, 23916, -31344, 34008, -45084, 44118, -44940, 53748, -95316, 125500, -119136, 103632, -100772, 156048, -238668, 332896, -334596, 293616, -253552, 335352, -591780, 833340, -789012, 572634, -678492, 998508, -1350228, 1632840, -1863108, 1749036, -1538120, 1841244, -2950512, 3959160, -3671160, 3113532, -3879628, 5512488, -6007836, 6159684, -7625868, 9420576, -8502888, 7725780, -12080952, 16859826, -14906736, 12391572, -18666168, 26180532, -23512824, 18486132, -29125692, 43754556
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OFFSET
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0,2
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COMMENTS
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Compare to the coefficient of y^0 in G(x,y)^2, which equals theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 +...+ 2*(-x)^(n^2) +..., where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n.
Note the curious identity: Sum_{n=-oo..+oo} (1-x^n)^n * x^n = 0.
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 - 12*x + 18*x^2 + 20*x^3 + 24*x^4 - 96*x^5 - 4*x^6 - 120*x^7 + 138*x^8 - 116*x^9 + 456*x^10 - 132*x^11 + 356*x^12 - 540*x^13 + 900*x^14 +...
such that A(x) is the coefficient of y^0 in G(x,y)^4 where
G(x,y) = N(x,y) + P(x,y), with
P(x,y) = 1 + x*y*(1-x) + (x*y)^2*(1-x^2)^2 + (x*y)^3*(1-x^3)^3 + (x*y)^4*(1-x^4)^4 + (x*y)^5*(1-x^5)^5 + (x*y)^6*(1-x^6)^6 +...+ (x*y)^n*(1-x^n)^n +...
N(x,y) = (-1/y)/(1-x) + (x/y)^2/(1-x^2)^2 + (-x^2/y)^3/(1-x^3)^3 + (x^3/y)^4/(1-x^4)^4 + (-x^4/y)^5/(1-x^5)^5 +...+ (-x^(n-1)/y)^n/(1-x^n)^n +...
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PROG
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(PARI) {a(n) = my(A=sum(m=-sqrtint(n)-1, n+1, x^m*(1-x^m)^m*y^m +x*O(x^n))); polcoeff(polcoeff(A^4, 0, y), n, x)}
for(n=0, 80, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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