A128713 Expansion of q^(-3/8)* eta(q)^7* eta(q^4)^2/ eta(q^2)^3 in powers of q.

1, -7, 17, -14, 0, -7, 2, 41, -31, 25, -79, 0, 35, 89, 0, -46, -31, -103, 49, 0, 161, -85, 17, -14, 0, 0, 113, -142, -223, 0, 115, 233, 0, 146, -175, 41, -94, 0, -271, 0, 34, -7, 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

Offset

0,2

Links

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

Formula

Euler transform of period 4 sequence [ -7, -4, -7, -6, ...].

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

Example

q^3 - 7*q^11 + 17*q^19 - 14*q^27 - 7*q^43 + 2*q^51 + 41*q^59 - 31*q^67 + ...

Mathematica

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 *)
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 *)

Prog

(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)))))}
(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))}

Crossrefs

Cf. A128711(4n+1)= 2*a(n). A030207(8n+3) = -2*a(n).

Sequence in context: A107804 A276809 A274916 * A283163 A196164 A071615

Adjacent sequences: A128710 A128711 A128712 * A128714 A128715 A128716

Keyword

sign

Author

Michael Somos, Mar 24 2007

Status

approved