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A113260
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Expansion of (-1 + psi(q)^5/psi(q^5) - 25q^2 psi(q)*psi(q^5)^3)/5 in powers of q where psi(q) is a Ramanujan theta function.
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1
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1, -3, -2, 5, 1, 6, -6, -11, 7, -3, 12, -10, -12, 18, -2, 21, -16, -21, 20, 5, 12, -36, -22, 22, 1, 36, -20, -30, 30, 6, 32, -43, -24, 48, -6, 35, -36, -60, 24, -11, 42, -36, -42, 60, 7, 66, -46, -42, 43, -3, 32, -60, -52, 60, 12, 66, -40, -90, 60, -10, 62, -96, -42, 85, -12, 72, -66, -80, 44, 18, 72, -77, -72, 108
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OFFSET
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1,2
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COMMENTS
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REFERENCES
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Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 249, Entry 8(iv).
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LINKS
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FORMULA
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a(n) is multiplicative with a(2^e) = ((-2)^(e+2)-1)/3, a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 4 (mod 5), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2, 3 (mod 5).
G.f.: Sum_{k>0} (k*x^k / (1 + x^k)) * Kronecker(5, k).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(10*sqrt(5)) = 0.441382... . - Amiram Eldar, Jan 22 2024
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EXAMPLE
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G.f. = x - 3*x^2 - 2*x^3 + 5*x^4 + x^5 + 6*x^6 - 6*x^7 - 11*x^8 + 7*x^9 + ... - Michael Somos, Sep 07 2018
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, -DivisorSum[ n, # KroneckerSymbol[ 5, #] (-1)^(n/#) &]]; (* Michael Somos, Sep 07 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, -sumdiv(n, d, d * kronecker(5, d) * (-1)^(n/d)))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p==2, ((-2)^(e+2) - 1)/3, p *= kronecker(5, p); (p^(e+1) - 1) / (p - 1))))};
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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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