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A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.

%I A122861 #15 Feb 16 2025 08:33:02
%S A122861 1,-3,2,0,2,-3,2,0,1,-6,2,0,2,0,2,0,3,-6,0,0,2,-3,2,0,2,-6,2,0,0,0,4,
%T A122861 0,2,-3,2,0,2,-6,0,0,1,-6,2,0,4,0,2,0,0,-6,2,0,2,0,2,0,3,-6,2,0,2,0,0,
%U A122861 0,2,-9,2,0,0,-6,2,0,4,0,2,0,2,0,0,0,2,-6,4,0,0,-3,4,0,0,-6,2,0,2,0,2,0,1,-6,0,0,4,-6,2,0,2
%N A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.
%C A122861 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).
%H A122861 G. C. Greubel, <a href="/A122861/b122861.txt">Table of n, a(n) for n = 0..1000</a>
%H A122861 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A122861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A122861 Expansion of q^(-1/3)*eta(q)^3*eta(q^6)^2/(eta(q^2)^2*eta(q^3)) in powers of q.
%F A122861 Euler transform of period 6 sequence [ -3, -1, -2, -1, -3, -2, ...].
%F A122861 a(n) = b(3n+1) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
%F A122861 a(4n+3) = 0.
%F A122861 a(n) = A115979(3n+1) = A097109(3n+1).
%F A122861 a(2n) = A097195(n) = A033687(2n); a(2n+1) = -3*A033687(2n+1).
%F A122861 a(n) = (-1)^n * A129576(n). - _Amiram Eldar_, Jan 28 2024
%t A122861 A122861[n_] := SeriesCoefficient[(QPochhammer[q]^3*QPochhammer[q^6]^2)/(QPochhammer[q^2]^2 *QPochhammer[q^3]), {q, 0, n}]; Table[A122861[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 05 2017 *)
%o A122861 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^2/eta(x^2+A)^2/eta(x^3+A), n))}
%o A122861 (PARI) {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if(p==3, 0, if(p%6==1, e+1, !(e%2)))))))}
%Y A122861 Cf. A033687, A097109, A097195, A115979, A129576.
%Y A122861 Cf. A000122, A000700, A010054, A121373.
%K A122861 sign
%O A122861 0,2
%A A122861 _Michael Somos_, Sep 15 2006