0,2

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

G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(4^n*x) )^n/n!.

G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (4^n*x)^k/(1 - (4^n*x)^k) )/k ]^n/n!.

a(n) = [x^n] P(x)^(4^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).

G.f.: A(x) = 1 + 4*x + 152*x^2 + 49920*x^3 + 191701440*x^4 +...

A(x) = 1 - log(eta(4*x)) + log(eta(16*x))^2/2! - log(eta(64*x))^3/3! +-...

...

Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:

P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...

then a(n) is the coefficient of x^n in P(x)^(4^n):

P(x)^(4^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];

P(x)^(4^1): [1,(4),14,40,105,252,574,1240,2580,5180,10108,...];

P(x)^(4^2): [1,16,(152),1088,6460,33440,155584,663936,2636326,...];

P(x)^(4^3): [1,64,2144,(49920),905840,13627264,176638592,...];

P(x)^(4^4): [1,256,33152,2894848,(191701440),10266643968,...];

P(x)^(4^5): [1,1024,525824,180531200,46620870400,(9659304851456),...];

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

(PARI) {a(n)=polcoeff(1/eta(x+x*O(x^n))^(4^n), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, (-1)^m*log(eta(4^m*x+x*O(x^n)))^m/m!), n)}

(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)}

Cf. A000041, A158102, A158103, A158105, A158112, A158113, A158114, A158115.

Sequence in context: A159197 A160470 A006439 this_sequence A093977 A003736 A104133

Adjacent sequences: A158101 A158102 A158103 this_sequence A158105 A158106 A158107

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 12 2009