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A230277
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Expansion of f(-q^6)^6 * ( q * chi(-q^3)^4 + 2*q^2 / chi(-q^3)^4 ) in powers of q where chi(), f() are Ramanujan theta functions.
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3
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1, 2, 0, -4, 8, 0, 0, 8, 0, 16, 0, 0, -10, 0, 0, -16, -16, 0, 0, -32, 0, 0, 0, 0, 39, -20, 0, 0, -40, 0, 0, 32, 0, -32, 0, 0, -70, 0, 0, 64, 80, 0, 0, 0, 0, 0, 0, 0, 49, 78, 0, 40, 56, 0, 0, 0, 0, -80, 0, 0, -22, 0, 0, -64, -80, 0, 0, 64, 0, 0, 0, 0, 110
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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 (eta(q^3)^2 * eta(q^6))^2 + 2 * (eta(q^6)^5 / eta(q^3)^2)^2 in powers of q.
a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(3^e) = 0^e, a(p^e) = p^e * (1 + (-1)^e)/2 if p == 3, 7 (mod 8), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) and a(p) = t * (-1)^(t mod 3) if p == 1, 5 (mod 8) where t = 2 * if( p == 5 (mod 6) then 4*x*y else p - 8*y^2 ) and p = x^2 + 4*y^2.
a(2*n) = -2 * a(n). a(3*n) = a(4*n + 3) = 0.
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EXAMPLE
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G.f. = q + 2*q^2 - 4*q^4 + 8*q^5 + 8*q^8 + 16*q^10 - 10*q^13 - 16*q^16 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ With[{t = QPochhammer[ q^3, q^6]^4}, QPochhammer[ q^6]^6 (q t + 2 q^2 / t)], {q, 0, n}]
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A)^2 * eta(x^6 + A))^2 + 2 * x * (eta(x^6 + A)^5 / eta(x^3 + A)^2)^2, n))}
(PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor( n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==3, 0, if( p==2, -(-2)^e, if( p%8==3 || p%8==7, if(e%2, 0, p^e), for( i = 1, sqrtint( p\4), if( issquare( p - 4*i^2, &x), y = i; break)); y = 2 * if( p%6 == 5, 4*x*y, p - 8*y^2); a1 = y = if( y%3==2, y, -y); a0 = 1; for( i=2, e, x = y*a1 - p^2*a0; a0 = a1; a1 = x); a1))))))}
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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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