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Series reversion of x - 3*x^3.
3

%I #34 Feb 22 2024 20:12:59

%S 1,3,27,324,4455,66339,1041012,16953624,283848543,4855304025,

%T 84482290035,1490628232080,26607713942628,479621100042756,

%U 8718235759397880,159628084420459248,2941328850997439439,54501093415540789605,1014898739548854163185

%N Series reversion of x - 3*x^3.

%C Regular zeros in the reverted sequence have been left out: If y = x - 3*x^3, then x = y + 3*y^3 + 27*y^5 + 324*y^7 + 4455*y^9 + 66339*y^11 + ...

%C Number of lattice paths that do not go below the x-axis from (0,0) to (3n,0) using steps D(1,-1) and three types of U(1,2). - _David Scambler_, Jun 22 2013

%H Robert Israel, <a href="/A217363/b217363.txt">Table of n, a(n) for n = 1..769</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1211.3963">Series Expansion of Generalized Fresnel Integrals</a>, arXiv:1211.3963 [math.CA], 2012, App. A.

%F D-finite with recurrence (2*n-1)*(2*n-2)*a(n) - 9*(3*n-4)*(3*n-5)*a(n-1) = 0.

%F a(n) = 3^(n-1)*A001764(n-1).

%F From _Benedict W. J. Irwin_, Jul 12 2016: (Start)

%F G.f.: (2/3)*sqrt(x)*sin(asin(9*sqrt(x)/2)/3).

%F E.g.f.: x*2F2(1/3,2/3;3/2,2;81*x/4).

%F (End)

%F a(n) ~ 3^(4*n - 7/2)*4^(-n)*n^(-3/2)/sqrt(Pi). - _Ilya Gutkovskiy_, Jul 12 2016

%p f:= k -> (3*k-3)!*3^(k-1)/(k-1)!/(2*k-1)!:

%p map(f, [$1..30]); # _Robert Israel_, May 07 2017

%t CoefficientList[Series[2/3 Sqrt[z] Sin[ArcSin[(9 Sqrt[z])/2]/3], {z, 0, 20}], z](* _Benedict W. J. Irwin_, Jul 12 2016 *)

%Y Cf. A005159 (revert x-3*x^2), A153231 (revert x-2*x^3).

%K nonn,easy

%O 1,2

%A _R. J. Mathar_, Oct 01 2012