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

1, 3, 27, 324, 4455, 66339, 1041012, 16953624, 283848543, 4855304025, 84482290035, 1490628232080, 26607713942628, 479621100042756, 8718235759397880, 159628084420459248, 2941328850997439439, 54501093415540789605, 1014898739548854163185

Offset: 1

Author

R. J. Mathar, Oct 01 2012

Keywords

• nonn
• easy

Comments

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 + ...

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

Links

• Robert Israel, Table of n, a(n) for n = 1..769
• R. J. Mathar, Series Expansion of Generalized Fresnel Integrals, arXiv:1211.3963 [math.CA], 2012, App. A.

Crossrefs

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

Programs

Maple

f:= k -> (3*k-3)!*3^(k-1)/(k-1)!/(2*k-1)!:
map(f, [$1..30]); # Robert Israel, May 07 2017

Mathematica

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

Formula

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

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

From Benedict W. J. Irwin, Jul 12 2016: (Start)

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

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

a(n) ~ 3^(4*n - 7/2)*4^(-n)*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016