cp's OEIS Frontend

A158112 a(n) = [x^n] eta(x)^(2^n).

%I A158112 #5 Nov 24 2015 03:27:38
%S A158112 1,-2,2,0,260,-72384,40618368,-64586112000,322969883412000,
%T A158112 -5357637135507147264,305435344239722874022912,
%U A158112 -61394251001333873555321724928,44347756583930343994689166159720448
%N A158112 a(n) = [x^n] eta(x)^(2^n).
%C A158112 Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
%F A158112 G.f.: A(x) = Sum_{n>=0} log( eta(2^n*x) )^n/n!.
%F A158112 G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (2^n*x)^k/(1 - (2^n*x)^k) )/k ]^n/n!.
%F A158112 a(n) = [x^n] Product_{k>=1} (1-x^k)^(2^n).
%e A158112 G.f.: A(x) = 1 - 2*x + 2*x^2 + 260*x^4 - 72384*x^5 +...
%e A158112 A(x) = 1 + log(eta(2*x)) + log(eta(4*x))^2/2! + log(eta(8*x))^3/3! +...
%e A158112 ...
%e A158112 Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
%e A158112 then a(n) is the coefficient of x^n in eta(x)^(2^n):
%e A158112 eta(x)^(2^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..];
%e A158112 eta(x)^(2^1): [1,(-2),-1,2,1,2,-2,0,-2,-2,1,0,0,2,3,-2,2,...];
%e A158112 eta(x)^(2^2): [1,-4,(2),8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...];
%e A158112 eta(x)^(2^3): [1,-8,20,(0),-70,64,56,0,-125,-160,308,0,110,0,...];
%e A158112 eta(x)^(2^4): [1,-16,104,-320,(260),1248,-3712,1664,6890,...];
%e A158112 eta(x)^(2^5): [1,-32,464,-3968,21576,(-72384),109120,215296,...];
%e A158112 eta(x)^(2^6): [1,-64,1952,-37632,512400,-5207936,(40618368),...]; ...
%e A158112 where terms in parenthesis form the initial terms of this sequence.
%t A158112 a[n_] := SeriesCoefficient[QPochhammer[q]^(2^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 24 2015 *)
%o A158112 (PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(2^n), n)}
%o A158112 (PARI) {a(n)=polcoeff(sum(m=0,n,log(eta(2^m*x+x*O(x^n)))^m/m!), n)}
%o A158112 (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(2^m*x)^k/(1-(2^m*x)^k)/k+x*O(x^n))^m/m!),n)}
%Y A158112 Cf. A010815, A158113, A158114, A158115, A158102, A158103, A158104, A158105.
%K A158112 sign
%O A158112 0,2
%A A158112 _Paul D. Hanna_, Mar 12 2009