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(5^n*x) )^n/n!.

G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (5^n*x)^k/(1 - (5^n*x)^k) )/k ]^n/n!.

a(n) = [x^n] P(x)^(5^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 + 5*x + 350*x^2 + 349125*x^3 + 6541895625*x^4 +...

A(x) = 1 - log(eta(5*x)) + log(eta(25*x))^2/2! - log(eta(125*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)^(5^n):

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

P(x)^(5^1): [1,(5),20,65,190,506,1265,2990,6765,14725,31027,...];

P(x)^(5^2): [1,25,(350),3575,29575,209405,1312675,7452225,...];

P(x)^(5^3): [1,125,8000,(349125),11676000,318906400,...];

P(x)^(5^4): [1,625,196250,41276875,(6541895625),833314453875,...];

P(x)^(5^5): [1,3125,4887500,5100915625,3996555181250,(2507423437503750),..];

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

(PARI) {a(n)=polcoeff(1/eta(x+x*O(x^n))^(5^n), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, (-1)^m*log(eta(5^m*x+x*O(x^n)))^m/m!), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, (5^m*x)^k/(1-(5^m*x)^k)/k+x*O(x^n))^m/m!), n)}

Cf. A000041, A158102, A158103, A158104, A158112, A158113, A158114, A158115.

Sequence in context: A086900 A124477 A059839 this_sequence A006108 A061456 A006430

Adjacent sequences: A158102 A158103 A158104 this_sequence A158106 A158107 A158108

nonn

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