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A176025 Series reversion of eta(-x) - 1. 5

%I

%S 1,1,2,5,15,49,169,603,2205,8217,31095,119185,461790,1805810,7117865,

%T 28250549,112806534,452862663,1826705940,7399893522,30092189864,

%U 122799412699,502709227763,2063939448400,8496355807149,35061664792175

%N Series reversion of eta(-x) - 1.

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

%H Vaclav Kotesovec, <a href="/A176025/b176025.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) satisfies: eta(-A(x)) = 1 + x, or, equivalently:

%F x = Sum_{n>=1} (-1)^[n/2] * A(x)^(n(3n-1)/2) * (1 + (-A(x))^n).

%F a(n) ~ c * d^n / n^(3/2), where d = 4.37926411884088478340484205014088510... and c = 0.422672515444252849172886523421828... - _Vaclav Kotesovec_, Nov 11 2017

%e G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 +...

%e eta(-x)-1 = x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 +...

%e eta(-x)-1 = Sum_{n>=1} (-1)^[n/2]*x^(n(3n-1)/2)*(1 + (-x)^n).

%t -InverseSeries[Series[QPochhammer[x], {x, 0, 20}]][[3]] (* _Vladimir Reshetnikov_, Nov 21 2015 *)

%o (PARI) a(n)=polcoeff(serreverse(-1+eta(-x+x*O(x^n))),n)

%Y Cf. A010815.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 06 2010

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Last modified January 22 13:21 EST 2021. Contains 340362 sequences. (Running on oeis4.)