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A247067
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Glaisher's chi_12(n).
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3
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1, -64, 0, 4096, 23506, 0, 0, -262144, 531441, -1504384, 0, 0, 6911282, 0, 0, 16777216, -47295038, -34012224, 0, 96280576, 0, 0, 0, 0, 308391411, -442322048, 0, 0, -173439758, 0, 0, -1073741824, 0, 3026882432, 0, 2176782336, -2050092718, 0, 0, -6161956864
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q * f(-x^2)^24 * phi(q)^2 * (chi(q)^24 - 92*q + 16*q^2 / chi(q)^24) in powers of q where phi(), chi(), f() are Ramanujan theta functions.
a(n) is multiplicative with a(2^e) = (-64)^e, a(p^e) = p^(6*e) * (1 + (-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p) * a(p^(e-1)) - p^12 * a(p^(e-2)) if p == 1 (mod 4) where a(p) = 2 * Re( (x + i*y)^12 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^13 (t/i)^13 f(t) where q = exp(2 Pi i t).
G.f.: ( Sum_{j,k in Z} (j + i*k)^12 * x^(j^2 + k^2) ) / 4, where i^2 = -1.
a(2*n) = (-4)^3 * a(n). a(4*n + 3) = 0.
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EXAMPLE
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G.f. = q - 64*q^2 + 4096*q^4 + 23506*q^5 - 262144*q^8 + 531441*q^9 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q EllipticTheta[3, 0, q]^2 QPochhammer[ q^2]^24 (QPochhammer[ -q, q^2]^24 - 92 q + 16 q^2/QPochhammer[ -q, q^2]^24), {q, 0, n}];
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PROG
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(PARI) {a(n) = my(r); if( n<1, 0, r = sqrtint(n); sum( x=-r, r, sum( y=-r, r, if( x^2 + y^2 == n, (x + I*y)^12) )) / 4 ) };
(PARI) {a(n) = local(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e = A[k, 2]; if( p==2, (-64)^e, if( p%4 == 3, if( e%2, 0, p^(6*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^12 ) * 2; for( i=2, e, y = x*a1 - p^12*a0; a0=a1; a1=y); a1))))) }; /* Michael Somos, Nov 18 2014 */
(Magma) A := Basis( CuspForms( Gamma1(4), 13), 40); A[1] - 64*A[2] + 4096*A[4] + 23506*A[5];
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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