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A121362 Expansion of eta(q)*eta(q^6)*eta(q^10)*eta(q^15)/(eta(q^3)*eta(q^5)) in powers of q.

%I A121362 #13 Feb 16 2025 08:33:02
%S A121362 1,-1,-1,1,-1,1,0,-1,1,1,0,-1,0,0,1,1,-2,-1,2,-1,0,0,-2,1,1,0,-1,0,0,
%T A121362 -1,2,-1,0,2,0,1,0,-2,0,1,0,0,0,0,-1,2,-2,-1,1,-1,2,0,-2,1,0,0,-2,0,0,
%U A121362 1,2,-2,0,1,0,0,0,-2,2,0,0,-1,0,0,-1,2,0,0,2,-1,1,0,-2,0,2,0,0,0,0,1,0,-2,-2,2,-2,1,0,-1,0,1,0,-2,0,0,0
%N A121362 Expansion of eta(q)*eta(q^6)*eta(q^10)*eta(q^15)/(eta(q^3)*eta(q^5)) in powers of q.
%C A121362 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).
%H A121362 G. C. Greubel, <a href="/A121362/b121362.txt">Table of n, a(n) for n = 1..1000</a>
%H A121362 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A121362 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A121362 Euler transform of period 30 sequence [ -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -2, ...].
%F A121362 Expansion of q*f(-q)f(-q^15)/(chi(-q^3)chi(-q^5)) in powers of q where f(),chi() are Ramanujan theta functions.
%F A121362 G.f.: x Product_{n>0} (1-x^n)(1+x^(3n))(1+x^(5n))(1-x^(15n)).
%F A121362 a(n) is multiplicative with a(2^e)=a(3^e)=a(5^e)=(-1)^e, a(p^e) = e+1 if p == 1,4 (mod 15), a(p^e) = (-1)^e*(e+1) if p == 2,8 (mod 15), a(p^e) = (1+( -1)^e)/2 if p == 7,11,13,14 (mod 15).
%t A121362 eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^6]*eta[q^10]*eta[q^15]/(eta[q^3]*eta[q^5]), {q, 0, n}]; Table[a[n], {n,1,50}] (* _G. C. Greubel_, Feb 11 2018 *)
%o A121362 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)*eta(x^10+A)*eta(x^15+A)/(eta(x^3+A)*eta(x^5+A)), n))}
%Y A121362 Cf. A082451(n) = |a(n)|.
%K A121362 sign,mult
%O A121362 1,17
%A A121362 _Michael Somos_, Jul 22 2006