cp's OEIS Frontend

A118271 Expansion of (9 * theta_4(q^3)^4 - theta_4(q)^4) / 8 in powers of q.

%I A118271 #17 Oct 28 2023 04:04:35
%S A118271 1,1,-3,-5,-3,6,15,8,-3,-23,-18,12,15,14,-24,-30,-3,18,69,20,-18,-40,
%T A118271 -36,24,15,31,-42,-77,-24,30,90,32,-3,-60,-54,48,69,38,-60,-70,-18,42,
%U A118271 120,44,-36,-138,-72,48,15,57,-93,-90,-42,54,231,72,-24,-100,-90,60,90,62,-96,-184,-3,84,180,68,-54,-120,-144
%N A118271 Expansion of (9 * theta_4(q^3)^4 - theta_4(q)^4) / 8 in powers of q.
%H A118271 G. C. Greubel, <a href="/A118271/b118271.txt">Table of n, a(n) for n = 0..1500</a>
%F A118271 Expansion of eta(q^2)^5 * eta(q^3)^3 / (eta(q) * eta(q^6)^3) in powers of q.
%F A118271 Expansion of b(q^2) * (4*b(q^4) - b(q)) / 3 in powers of q where b() is a cubic AGM theta function.
%F A118271 Euler transform of period 6 sequence [ 1, -4, -2, -4, 1, -4, ...].
%F A118271 a(n) is multiplicative with a(2^e) = -3 if e>0, a(3^e) = 4 - 3^(e+1), a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
%F A118271 a(3*n) = A109506(3*n). a(3*n + 1) = A109506(3*n + 1). a(3*n + 2) = -3 * A118272(n).
%F A118271 Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 2^(2-s)) * (1 - 2^(1-s)) * (1 - 3^(2-s)). - _Amiram Eldar_, Oct 28 2023
%e A118271 1 + q - 3*q^2 - 5*q^3 - 3*q^4 + 6*q^5 + 15*q^6 + 8*q^7 - 3*q^8 - ...
%t A118271 eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[eta[q^2]^5 *eta[q^3]^3/(eta[q]*eta[q^6]^3), {q, 0, 55}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jul 11 2018 *)
%o A118271 (PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * (-x)^k^2, 1 + x * O(x^n))^4 - 9 * sum( k=1, sqrtint(n\3), 2 * (-x^3)^k^2, 1 + x * O(x^n))^4, n) / -8)}
%o A118271 (PARI) {a(n) = if( n<1, n==0, -(-1)^n * ( sumdiv( n, d, d * (1 - if( d%3==0, 3) - if( d%4==0, 1) + if(d%12==0, 3)))))}
%o A118271 (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -3, if( p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1))))))}
%o A118271 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^3 / eta(x + A) / eta(x^6 + A)^3, n))}
%Y A118271 Cf. A109506, A118272.
%K A118271 sign,mult
%O A118271 0,3
%A A118271 _Michael Somos_, Apr 21 2006