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A113186
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Expansion of (25phi(q)phi^3(q^5)-phi^5(q)/phi(q^5)-24)/40 in powers of q where phi(q) is a Ramanujan theta function.
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1
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1, -1, -2, -1, 1, 2, -6, -1, 7, -1, 12, 2, -12, 6, -2, -1, -16, -7, 20, -1, 12, -12, -22, 2, 1, 12, -20, 6, 30, 2, 32, -1, -24, 16, -6, -7, -36, -20, 24, -1, 42, -12, -42, -12, 7, 22, -46, 2, 43, -1, 32, 12, -52, 20, 12, 6, -40, -30, 60, 2, 62, -32, -42, -1, -12, 24, -66, 16, 44, 6, 72, -7, -72, 36, -2, -20, -72, -24, 80, -1
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OFFSET
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1,3
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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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(iii).
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LINKS
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FORMULA
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a(n) is multiplicative with a(2^e) = -1 if e>0, a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 9 (mod 10), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 7 (mod 10).
G.f.: (25phi(q) phi(q^5)^3 - phi(q)^5/phi(q^5)-24)/40 where phi(q) = 1 + 2(q+q^4+q^9+...).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(18*sqrt(5)) = 0.245212... . - Amiram Eldar, Jan 28 2024
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MATHEMATICA
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a[n_]:= SeriesCoefficient[(25*EllipticTheta[3, 0, q]*(EllipticTheta[3, 0, q^5])^3 - (EllipticTheta[3, 0, q])^5/EllipticTheta[3, 0, q^5] - 24)/40, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 07 2018 *)
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PROG
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(PARI) a(n)=if(n<1, 0, (-1)^n*sumdiv(n, d, kronecker(20, d)*d*(-1)^d))
(PARI) {a(n)=local(A, p, e); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -1, if(p==5, 1, 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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