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A124449
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Expansion of (phi(-q^3)^4 - phi(-q)^4)/8 in powers of q where phi() is a Ramanujan theta function.
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4
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1, -3, 3, -3, 6, -9, 8, -3, 9, -18, 12, -9, 14, -24, 18, -3, 18, -27, 20, -18, 24, -36, 24, -9, 31, -42, 27, -24, 30, -54, 32, -3, 36, -54, 48, -27, 38, -60, 42, -18, 42, -72, 44, -36, 54, -72, 48, -9, 57, -93, 54, -42, 54, -81, 72, -24, 60, -90, 60, -54, 62, -96, 72, -3, 84, -108, 68, -54, 72, -144, 72, -27, 74, -114
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OFFSET
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1,2
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COMMENTS
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Ramanujan theta functions: f(q) := Product_{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) := Product_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
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FORMULA
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Expansion of eta(q)^3*eta(q^6)^5/(eta(q^2)^3*eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, -4, ...].
a(n) is multiplicative with a(2^e) = -3 if e>0, a(3^e) = 3^e, a(p^e) =(p^(e+1)-1)/(p-1) if p>3.
G.f.: x*Product_{k>0} (1+x^(3k))*(1-x^(6k))^4/(1+x^k)^3.
a(3*n) = 3*a(n). a(4*n) = a(2*n).
Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * (1 - 1/3^s) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023
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MATHEMATICA
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a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q^3]^4 - EllipticTheta[ 3, 0, -q]^4)/8, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 16 2017 *)
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PROG
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(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -3, if(p==3, 3^e, (p^(e+1)-1)/(p-1))))))}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^5/ eta(x^2+A)^3/eta(x^3+A), n))}
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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