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A133574
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Expansion of (5 * phi(q^5)^2 - phi(q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
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2
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1, -1, -1, 0, -1, 3, 0, 0, -1, -1, 3, 0, 0, -2, 0, 0, -1, -2, -1, 0, 3, 0, 0, 0, 0, 7, -2, 0, 0, -2, 0, 0, -1, 0, -2, 0, -1, -2, 0, 0, 3, -2, 0, 0, 0, 3, 0, 0, 0, -1, 7, 0, -2, -2, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 6, 0, 0, -2, 0, 0, 0, -1, -2, -2, 0, 0, 0
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OFFSET
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0,6
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COMMENTS
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LINKS
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FORMULA
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Expansion of psi(-q)^2 * chi(q) * chi(q^5) in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^10)^2 / (eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, -1, -2, -1, -1, 0, -2, -1, -1, -1, -2, ...].
Moebius transform is period 20 sequence [ -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4), A(x^8)) where f(u1, u2, u4, u8) = (u1 - u2)^2 * (u4 - 2*u8)^2 - u2 * u4 * (u2 - u4) * (u2 - 2*u4).
a(n) = -b(n) where b() is multiplicative with b(2^e) = 1, b(5^e) = 1-4*e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 10 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053694.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).
G.f.: 1 - (Sum_{k>0} x^k / (1 + x^(2*k)) - 5 * x^(5*k) / (1 + x^(10*k))).
Sum_{k=1..n} abs(a(k)) ~ (8*Pi/25) * n. - Amiram Eldar, Jan 27 2024
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EXAMPLE
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G.f. = 1 - q - q^2 - q^4 + 3*q^5 - q^8 - q^9 + 3*q^10 - 2*q^13 - q^16 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 3, 0, q^5]^2 - EllipticTheta[ 3, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ -q^5, q^10] / QPochhammer[ -q, q^2], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/4) EllipticTheta[ 2, Pi/4, x^(1/2)]^2 QPochhammer[ -x, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := If[n < 1, Boole[n == 0], DivisorSum[ n, If[Mod[#, 5] == 0, 5 KroneckerSymbol[-4, #/5], 0] - KroneckerSymbol[-4, #] &]]; (* Michael Somos, Oct 31 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, if( d%5==0, kronecker( -4, d/5) * 5) - kronecker( -4, d)))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod(k = 1, matsize(A) [1], [p, e] = A[k, ]; if(p == 2, 1, p == 5, 1 - 4*e, p%4 == 1, e+1, 1-e%2 )))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^5 + A) * eta(x^20 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(20), 1), 78); A[1] - A[2] - A[3] - A[5] + 3*A[6] - A[9]; /* Michael Somos, Oct 31 2015 /;
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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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