A158113 - OEIS

%I #6 Nov 24 2015 04:37:59

%S 1,-3,27,-2223,1411020,-6214529628,195816874969170,

%T -46124406363148894638,84075118460456018326675380,

%U -1215272486750697404426951511303060

%N a(n) = [x^n] eta(x)^(3^n).

%C Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

%F G.f.: A(x) = Sum_{n>=0} log( eta(3^n*x) )^n/n!.

%F G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (3^n*x)^k/(1 - (3^n*x)^k) )/k ]^n/n!.

%F a(n) = [x^n] Product_{k>=1} (1-x^k)^(3^n).

%e G.f.: A(x) = 1 - 3*x + 27*x^2 - 2223*x^3 + 1411020*x^4 +...

%e A(x) = 1 + log(eta(3*x)) + log(eta(9*x))^2/2! + log(eta(27*x))^3/3! +...

%e ...

%e Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...

%e then a(n) is the coefficient of x^n in eta(x)^(3^n):

%e eta(x)^(3^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,..];

%e eta(x)^(3^1): [1,(-3),0,5,0,0,-7,0,0,0,9,0,0,0,0,-11,0,0,0,0,0,13,..];

%e eta(x)^(3^2): [1,-9,(27),-12,-90,135,54,-99,-189,-85,657,-162,...];

%e eta(x)^(3^3): [1,-27,324,(-2223),9126,-19278,-5967,159030,...];

%e eta(x)^(3^4): [1,-81,3159,-78840,(1411020),-19222515,206322876,...];

%e eta(x)^(3^5): [1,-243,29160,-2303235,134665740,(-6214529628),...]; ...

%e where terms in parenthesis form the initial terms of this sequence.

%t a[n_] := SeriesCoefficient[QPochhammer[q]^(3^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 24 2015 *)

%o (PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(3^n), n)}

%o (PARI) {a(n)=polcoeff(sum(m=0,n,log(eta(3^m*x+x*O(x^n)))^m/m!), n)}

%o (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(3^m*x)^k/(1-(3^m*x)^k)/k+x*O(x^n))^m/m!),n)}

%Y Cf. A010815, A158112, A158114, A158115, A158102, A158103, A158104, A158105.

%K sign

%O 0,2

%A _Paul D. Hanna_, Mar 12 2009