%I #9 Jul 04 2018 12:43:52
%S 1,-7,17,-14,0,-7,2,41,-31,25,-79,0,35,89,0,-46,-31,-103,49,0,161,-85,
%T 17,-14,0,0,113,-142,-223,0,115,233,0,146,-175,41,-94,0,-271,0,34,-7,
%U 98,329,0,75,0,-343,35,0,0,-238,257,0,0,-439,322,-28,17,425,0,-391,401,169,0,-199,-205,-343,-511
%N Expansion of q^(-3/8)* eta(q)^7* eta(q^4)^2/ eta(q^2)^3 in powers of q.
%H G. C. Greubel, <a href="/A128713/b128713.txt">Table of n, a(n) for n = 0..2500</a>
%F Euler transform of period 4 sequence [ -7, -4, -7, -6, ...].
%F G.f.: Product_{k>0} (1-x^k)^6* (1+x^(2k))/ (1+x^k)^2.
%e q^3 - 7*q^11 + 17*q^19 - 14*q^27 - 7*q^43 + 2*q^51 + 41*q^59 - 31*q^67 + ...
%t QP = QPochhammer; s = QP[q]^7*(QP[q^4]^2/QP[q^2]^3) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[ q^(-3/8)* eta[q]^7*eta[q^4]^2/eta[q^2]^3, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o (PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<0, 0, n= 8*n+3; A=factor(n); 1/2*prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if( p%8>4, if(e%2, 0, p^e), for(i=1, sqrtint(p\2), if( issquare(p-2*i^2, &x), break)); a0=1; a1=y=2*(2*x^2 -p)* (-1)^((p-1)/2); for(i=2, e, x=y*a1-p^2*a0; a0=a1; a1=x); a1)))))}
%o (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^7* eta(x^4+A)^2/ eta(x^2+A)^3, n))}
%Y Cf. A128711(4n+1)= 2*a(n). A030207(8n+3) = -2*a(n).
%K sign
%O 0,2
%A _Michael Somos_, Mar 24 2007