A158104 a(n) = [x^n] 1/eta(x)^(4^n).
1, 4, 152, 49920, 191701440, 9659304851456, 6631121047328399360, 63121566305846614746333184, 8450364710466604643357631528951808, 16116664948206611884835318695391740990586880
Offset: 0
Keywords
Examples
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.
Programs
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Mathematica
a[n_] := SeriesCoefficient[1/QPochhammer[q]^(4^n), {q, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Nov 24 2015 *) -
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)}
Formula
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).
Comments