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A143380
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Expansion of q^(-1/6) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)) in powers of q.
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4
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1, 2, 0, 0, 1, -2, 0, 0, -3, 0, 0, 0, -2, -2, 0, 0, 2, -2, 0, 0, -1, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, -2, 0, 0, 4, 2, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 2, -4, 0, 0, -1, -2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2, -2, 0, 0, -2, 2, 0, 0, 0
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of psi(x)^2 * chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 2, -3, 2, -2, ...].
a(n) = (-1)^(-n / 2) * b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 8) or p == 23 (mod 24), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3 (mod 8) or p == 17 (mod 24) and p>3, b(p^e) = (e+1) * s^e if p == 1, 7 (mod 24) where p = x^2 + 6*y^2 and s = Kronecker(12, x) * (-1)^((p-1) / 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 1152^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143378.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: Product_{k>0} (1 - (-x)^k)^2 * (1 + x^(2*k)).
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EXAMPLE
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G.f. = 1 + 2*x + x^4 - 2*x^5 - 3*x^8 - 2*x^12 - 2*x^13 + 2*x^16 - 2*x^17 + ...
G.f. = q + 2*q^7 + q^25 - 2*q^31 - 3*q^49 - 2*q^73 - 2*q^79 + 2*q^97 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^5 / (QPochhammer[ x]^2 QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)
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PROG
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(PARI) {a(n)= my(A, p, e, x); if(n<0, 0, A = factor(6*n + 1); simplify( I^-n * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e ))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + 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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