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

1, 1, 2, 5, 15, 49, 169, 603, 2205, 8217, 31095, 119185, 461790, 1805810, 7117865, 28250549, 112806534, 452862663, 1826705940, 7399893522, 30092189864, 122799412699, 502709227763, 2063939448400, 8496355807149, 35061664792175

Offset

0,3

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Formula

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

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

a(n) ~ c * d^n / n^(3/2), where d = 4.37926411884088478340484205014088510... and c = 0.422672515444252849172886523421828... - Vaclav Kotesovec, Nov 11 2017

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 +...
eta(-x)-1 = x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 +...
eta(-x)-1 = Sum_{n>=1} (-1)^[n/2]*x^(n(3n-1)/2)*(1 + (-x)^n).

Maple

# Using function CompInv from A357588.
CompInv(26, proc(n) 24*n + 1; if issqr(%) then sqrt(%); (-1)^(n + irem(iquo(% + irem(%, 6), 6), 2)) else 0 fi end); # Peter Luschny, Oct 05 2022

Mathematica

-InverseSeries[Series[QPochhammer[x], {x, 0, 20}]][[3]] (* Vladimir Reshetnikov, Nov 21 2015 *)

Prog

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

Crossrefs

Cf. A010815.

Sequence in context: A149938 A148365 A001892 * A084082 A190270 A364588

Adjacent sequences: A176022 A176023 A176024 * A176026 A176027 A176028

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2010

Status

approved