cp's OEIS Frontend

A128712 Expansion of q^(-1/8)* eta(q)^5* eta(q^2)^3/ eta(q^4)^2 in powers of q.

%I A128712 #9 Jul 04 2018 12:43:23
%S A128712 1,-5,2,25,-28,-46,49,68,0,-142,-11,146,-94,0,98,75,-28,-238,0,-10,0,
%T A128712 169,164,0,98,-124,-476,0,-125,434,194,-316,386,0,0,-238,-285,392,0,
%U A128712 -526,356,0,-478,0,194,795,230,0,0,-124,-766,-334,-412,50,578,-245,866,-238,0,196,0,644,0,0,-952,-1006
%N A128712 Expansion of q^(-1/8)* eta(q)^5* eta(q^2)^3/ eta(q^4)^2 in powers of q.
%H A128712 G. C. Greubel, <a href="/A128712/b128712.txt">Table of n, a(n) for n = 0..2500</a>
%F A128712 Euler transform of period 4 sequence [ -5, -8, -5, -6, ...].
%F A128712 G.f.: Product_{k>0} (1-x^k)^6* (1+x^k)/ (1+x^(2k))^2.
%e A128712 q - 5*q^9 + 2*q^17 + 25*q^25 - 28*q^33 - 46*q^41 + 49*q^49 + 68*q^57 + ...
%t A128712 QP = QPochhammer; s = QP[q]^5*(QP[q^2]^3/QP[q^4]^2) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%t A128712 eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/8)* eta[q]^5*eta[q^2]^3/eta[q^4]^2, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o A128712 (PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<0, 0, n= 8*n+1; A=factor(n); 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 A128712 (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^5* eta(x^2+A)^3/ eta(x^4+A)^2, n))}
%Y A128712 Cf. A128711(4n)= a(n). A030207(8n+1) = a(n).
%K A128712 sign
%O A128712 0,2
%A A128712 _Michael Somos_, Mar 24 2007