cp's OEIS Frontend

A158114 a(n) = [x^n] eta(x)^(4^n).

%I A158114 #2 Mar 30 2012 18:37:16
%S A158114 1,-4,104,-37632,166534720,-9109541173248,6487005386806124544,
%T A158114 -62637995710787181892993024,8428730138560436521519921925857280,
%U A158114 -16103390694987849639716307556519680725483520
%N A158114 a(n) = [x^n] eta(x)^(4^n).
%C A158114 Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
%F A158114 G.f.: A(x) = Sum_{n>=0} log( eta(4^n*x) )^n/n!.
%F A158114 G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (4^n*x)^k/(1 - (4^n*x)^k) )/k ]^n/n!.
%F A158114 a(n) = [x^n] Product_{k>=1} (1-x^k)^(4^n).
%e A158114 G.f.: A(x) = 1 - 4*x + 104*x^2 - 37632*x^3 + 166534720*x^4 +...
%e A158114 A(x) = 1 + log(eta(4*x)) + log(eta(16*x))^2/2! + log(eta(64*x))^3/3! +...
%e A158114 ...
%e A158114 Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
%e A158114 then a(n) is the coefficient of x^n in eta(x)^(4^n):
%e A158114 eta(x)^(4^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..];
%e A158114 eta(x)^(4^1): [1,(-4),2,8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...];
%e A158114 eta(x)^(4^2): [1,-16,(104),-320,260,1248,-3712,1664,6890,...];
%e A158114 eta(x)^(4^3): [1,-64,1952,(-37632),512400,-5207936,40618368,...];
%e A158114 eta(x)^(4^4): [1,-256,32384,-2698240,(166534720),-8118668800,...];
%e A158114 eta(x)^(4^5): [1,-1024,522752,-177385472,45010254080,(-9109541173248), ...];
%e A158114 where terms in parenthesis form the initial terms of this sequence.
%o A158114 (PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(4^n), n)}
%o A158114 (PARI) {a(n)=polcoeff(sum(m=0,n,log(eta(4^m*x+x*O(x^n)))^m/m!), n)}
%o A158114 (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(4^m*x)^k/(1-(4^m*x)^k)/k+x*O(x^n))^m/m!),n)}
%Y A158114 Cf. A010815, A158112, A158113, A158115, A158102, A158103, A158104, A158105.
%K A158114 sign
%O A158114 0,2
%A A158114 _Paul D. Hanna_, Mar 12 2009