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A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

%I A080965 #10 Nov 25 2015 02:14:44
%S A080965 1,4,2,-8,-4,8,-8,-16,6,12,8,-8,-8,24,0,-16,12,16,10,-24,-8,16,-24,
%T A080965 -16,8,28,8,-32,-16,8,0,-32,6,32,16,-16,-12,40,-24,-16,24,16,16,-40,
%U A080965 -8,40,0,-32,24,36,10,-16,-24,24,-32,-48,0,32,24,-24,-16,40,0,-48,12,16,16
%N A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.
%C A080965 Euler transform of period 4 sequence [4,-8,4,-3,...].
%H A080965 Alois P. Heinz, <a href="/A080965/b080965.txt">Table of n, a(n) for n = 0..10000</a>
%F A080965 G.f.: Product_{n>0} (1-x^(2n))^12/((1-x^n)^4(1-x^(4n))^5).
%p A080965 with(numtheory):
%p A080965 a:= proc(n) option remember; `if`(n=0, 1, add(add([-3, 4, -8, 4]
%p A080965       [1+irem(d, 4)]*d, d=divisors(j)) *a(n-j), j=1..n)/n)
%p A080965     end:
%p A080965 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 05 2015
%t A080965 a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, {-3, 4, -8, 4}[[1 + Mod[#, 4]]]*#&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 25 2015, after _Alois P. Heinz_ *)
%o A080965 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-4*eta(X^2)^12*eta(X^4)^-5,n))
%Y A080965 a(n)=A080964(4n)=2*A072071(4n)-A072070(4n).
%Y A080965 A083703(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
%Y A080965 a(2n)=0 iff A005875(n)=0.
%K A080965 sign
%O A080965 0,2
%A A080965 _Michael Somos_, Feb 28 2003