OFFSET
0,2
COMMENTS
Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
FORMULA
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).
EXAMPLE
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.
MATHEMATICA
a[n_] := SeriesCoefficient[1/QPochhammer[q]^(5^n), {q, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Nov 24 2015 *)
PROG
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2009
STATUS
approved