0,2

G. C. Greubel, Table of n, a(n) for n = 0..2500

Euler transform of period 4 sequence [ -5, -8, -5, -6, ...].

G.f.: Product_{k>0} (1-x^k)^6* (1+x^k)/ (1+x^(2k))^2.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(Pi / 8) * Pi^(3/2) * 2^(3/8) / Gamma(3/4)^6 = A388587. - Simon Plouffe, Sep 18 2025

q - 5*q^9 + 2*q^17 + 25*q^25 - 28*q^33 - 46*q^41 + 49*q^49 + 68*q^57 + ...

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 *)

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 *)

(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)))))}

(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))}

Cf. A128711(4n)= a(n). A030207(8n+1) = a(n).

Sequence in context: A104064 A038244 A135138 * A202141 A100080 A117734

Adjacent sequences: A128709 A128710 A128711 * A128713 A128714 A128715

Michael Somos, Mar 24 2007

approved