A138809 Expansion of 8 * eta(q)^7 / eta(q^7) + 49 * (eta(q) * eta(q^7))^3 in powers of q.

8, -7, -35, 56, -147, 168, 280, -7, -595, -511, 840, -854, 1176, 1176, -35, -1344, -2387, 2016, -2555, 2520, 3528, 56, -4270, -3710, 4760, -4207, 5880, 4592, -147, -5894, -6720, 6720, -9555, 6832, 10080, 168, -10731, -9590, 12600, -9408, 14280, 11760, 280, -12950, -17934

Offset

0,1

Links

Table of n, a(n) for n=0..44.

Formula

a(n) = -7 * b(n) where b(n) is multiplicative and b(7^e) = 1, ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1, 2, 4 (mod 7), (-(-p^2)^(e+1) + 1) / (p^2 + 1) if p == 3, 5, 6 (mod 7).

G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(7/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A105634.

Example

8 - 7*q - 35*q^2 + 56*q^3 - 147*q^4 + 168*q^5 + 280*q^6 - 7*q^7 - 595*q^8 + ...

Prog

(PARI) {a(n) = if( n<1, 8 * (n==0), -7 * sumdiv(n, d, d^2 * kronecker(-7, d)))}
(PARI) {a(n) = local(A, p, e); if( n<1, 8 * (n==0), A = factor(n); -7 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==7, 1, if(kronecker(p, 7)==1, ((p^2)^(e+1) - 1) / (p^2 - 1), (-(-p^2)^(e+1) + 1) / (p^2 + 1)))))) }
(PARI) {a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); polcoeff( if(B = eta(x^7 + A), A = eta(x + A); 49 * x * (A * B)^3 + 8 * A^7 / B), n))}

Crossrefs

Sequence in context: A267093 A090099 A365486 * A286460 A317231 A237646

Adjacent sequences: A138806 A138807 A138808 * A138810 A138811 A138812

Keyword

sign

Author

Michael Somos, Mar 31 2008

Status

approved