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%I #20 Mar 12 2021 22:24:44
%S 1,-1,0,-1,0,0,1,-1,1,0,2,0,0,-1,0,-1,0,-1,0,0,0,-2,2,0,1,0,0,-1,2,0,
%T 0,-1,0,0,0,-1,2,0,0,0,0,0,2,-2,0,-2,0,0,1,-1,0,0,2,0,0,-1,0,-2,0,0,0,
%U 0,1,-1,0,0,2,0,0,0,2,-1,0,-2,0,0,2,0,2,0,1,0,0,0,0,-2,0,-2,0,0,0,-2,0,0,0,0,0,-1,2,-1,0,0,0,0,0
%N Expansion of q * psi(-q) * psi(-q^7) in powers of q where psi(q) is a Ramanujan theta function.
%C 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 G. C. Greubel, <a href="/A121454/b121454.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of eta(q)*eta(q^4)*eta(q^7)*eta(q^28)/(eta(q^2)*eta(q^14)) in powers of q.
%F Euler transform of period 28 sequence [ -1, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, -1, -2, ...].
%F Moebius transform is period 28 sequence [ 1, -2, -1, 0, -1, 2, 0, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, 0, 0, -2, 1, 0, 1, 2, -1, 0, ...].
%F Multiplicative with a(2^e) = -1 if e>0, a(7^e) = 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).
%F G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^4*w*v +2*u^3*w*v^2 +2*u^2*w^2*v^2 +4*u^3*w^3 +4*u^3*w^2*v +8*u*w^4*v +8*u*w^3*v^2 +8*u^2*w^3*v -u^2*v^4 -2*u^2*w*v^3 -4*w^2*v^4 -4*u*w^2*v^3.
%F G.f.: Sum_{k>0} (-1)^k *x^k*(1-x^(2*k))*(1-x^(4*k))*(1-x^(6*k))/(1-x^(14*k)) = x * Product_{k>0} (1-x^k)*(1+x^(2*k))*(1-x^(7*k))*(1+ x^(14*k)).
%F a(7*n) = a(n). a(7*n+3) = a(7*n+5) = a(7*n+6) = 0.
%F a(n) = (-1)^(n+1)*A035162(n).
%t eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[eta[q] eta[q^4] eta[q^7] eta[q^28]/(eta[q^2] eta[q^14]), {q, 0, 100}], q]] (* _G. C. Greubel_, Apr 19 2018 *)
%o (PARI) {a(n)=if(n<1, 0, -(-1)^n*sumdiv(n,d,kronecker(-28,d)))}
%o (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)*eta(x^7+A)*eta(x^28+A)/eta(x^2+A)/eta(x^14+A), n))}
%Y Cf. A035162.
%K sign,mult
%O 1,11
%A _Michael Somos_, Jul 30 2006
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