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A217363 Series reversion of x - 3*x^3.

%I A217363 #51 Jan 23 2025 17:52:31
%S A217363 1,3,27,324,4455,66339,1041012,16953624,283848543,4855304025,
%T A217363 84482290035,1490628232080,26607713942628,479621100042756,
%U A217363 8718235759397880,159628084420459248,2941328850997439439,54501093415540789605,1014898739548854163185
%N A217363 Series reversion of x - 3*x^3.
%C A217363 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 A217363 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 A217363 Robert Israel, <a href="/A217363/b217363.txt">Table of n, a(n) for n = 1..769</a>
%H A217363 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 A217363 D-finite with recurrence (2*n-1)*(2*n-2)*a(n) - 9*(3*n-4)*(3*n-5)*a(n-1) = 0.
%F A217363 a(n) = 3^(n-1)*A001764(n-1).
%F A217363 From _Benedict W. J. Irwin_, Jul 12 2016: (Start)
%F A217363 G.f.: (2/3)*sqrt(x)*sin(asin(9*sqrt(x)/2)/3).
%F A217363 E.g.f.: x*2F2(1/3,2/3;3/2,2;81*x/4). (End)
%F A217363 a(n) ~ 3^(4*n - 7/2)*4^(-n)*n^(-3/2)/sqrt(Pi). - _Ilya Gutkovskiy_, Jul 12 2016
%p A217363 f:= k -> (3*k-3)!*3^(k-1)/(k-1)!/(2*k-1)!:
%p A217363 map(f, [$1..30]); # _Robert Israel_, May 07 2017
%t A217363 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 A217363 Cf. A005159 (revert x-3*x^2), A153231 (revert x-2*x^3).
%K A217363 nonn,easy
%O A217363 1,2
%A A217363 _R. J. Mathar_, Oct 01 2012