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A138809 Expansion of 8 * eta(q)^7 / eta(q^7) + 49 * (eta(q) * eta(q^7))^3 in powers of q. 2

%I

%S 8,-7,-35,56,-147,168,280,-7,-595,-511,840,-854,1176,1176,-35,-1344,

%T -2387,2016,-2555,2520,3528,56,-4270,-3710,4760,-4207,5880,4592,-147,

%U -5894,-6720,6720,-9555,6832,10080,168,-10731,-9590,12600,-9408,14280,11760,280,-12950,-17934

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

%F 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).

%F 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.

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

%o (PARI) {a(n) = if( n<1, 8 * (n==0), -7 * sumdiv(n, d, d^2 * kronecker(-7, d)))}

%o (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)))))) }

%o (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))}

%K sign

%O 0,1

%A _Michael Somos_, Mar 31 2008

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Last modified February 25 11:54 EST 2021. Contains 341606 sequences. (Running on oeis4.)