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A130539
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Expansion of q^(-1/3) * a(q) * b(q) * c(q) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.
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5
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1, 4, -13, 0, -1, 16, 11, 0, 25, -52, -46, 0, 47, 0, -22, 0, 120, -4, 0, 0, -121, 64, -109, 0, -97, 44, 131, 0, 0, 0, 13, 0, 167, 100, -37, 0, -214, -208, 0, 0, 121, -184, 146, 0, -143, 0, 251, 0, 0, 188, 59, 0, -118, 0, 299, 0, -168, -88, -325, 0, -313
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OFFSET
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0,2
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COMMENTS
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Denoted by g_3(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 27 form of weight 3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
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LINKS
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FORMULA
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Expansion of q^(-1/3) * ( eta(q)^5 * eta(q^3) + 9 * eta(q)^2 * eta(q^3) * eta(q^9)^3 ) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^e if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = x^2 - 2*p, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3).
G.f.: Sum_{k>=0} a(k) * x^(3*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 7*v*v) * x^(u*u + u*v + 7*v*v). - Michael Somos, Jun 14 2007
a(4*n + 1) = 4*a(n). a(4*n + 3) = 0. - Michael Somos, Oct 20 2015
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EXAMPLE
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G.f. = 1 + 4*x - 13*x^2 - x^4 + 16*x^5 + 11*x^6 + 25*x^8 - 52*x^9 - 46*x^10 + ...
G.f. = q + 4*q^4 - 13*q^7 - q^13 + 16*q^16 + 11*q^19 + 25*q^25 - 52*q^28 - ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^3] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3), {x, 0, n}]; (* Michael Somos, Oct 20 2015 *)
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PROG
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(PARI) {a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, if( p%3==2, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x=y*a1 - p^2*a0; a0=a1; a1=x); a1))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A) * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3), 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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