A107266 Expansion of (1-6*x-sqrt((1-6*x)^2-4*6*x^2))/(2*6*x^2).

1, 6, 42, 324, 2664, 22896, 203256, 1849392, 17156448, 161663040, 1543053888, 14887836288, 144963737856, 1422685140480, 14058304458624, 139754913276672, 1396721001457152, 14025182471414784, 141432971217841152, 1431708373864249344, 14543342842406252544, 148198801896234491904

Offset

0,2

Comments

Series reversion of x/(1+6x+6x^2). Transform of 6^n under the matrix A107131. A row of A107267.

Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 6 colors and D(1,-1) one color. - Paul Barry, May 16 2005

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: (1-6*x-sqrt(1-12*x+12*x^2))/(12*x^2).

a(n) = Sum_{k=0..n} 1/(k+1) * C(k+1,n-k+1) * C(n,k) * 6^k.

E.g.f.: a(n) = n! * [x^n] exp(6*x)*BesselI(1, 2*sqrt(6)*x)/(sqrt(6)*x). -Peter Luschny, Aug 25 2012

D-finite with recurrence: (n+2)*a(n) = 6*(2*n+1)*a(n-1) - 12*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(44+18*sqrt(6))*(6+2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

G.f.: 1/(1 - 6*x - 6*x^2/(1 - 6*x - 6*x^2/(1 - 6*x - 6*x^2/(1 - 6*x - 6*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

Mathematica

CoefficientList[Series[(1-6*x-Sqrt[1-12*x+12*x^2])/(12*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Prog

(PARI) x='x+O('x^66); Vec((1-6*x-sqrt(1-12*x+12*x^2))/(12*x^2)) \\ Joerg Arndt, May 15 2013

Crossrefs

Sequence in context: A244902 A153293 A145301 * A142985 A118351 A033296

Adjacent sequences: A107263 A107264 A107265 * A107267 A107268 A107269

Keyword

easy,nonn

Author

Paul Barry, May 15 2005

Status

approved