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%I #25 Jan 30 2020 21:29:15
%S 1,6,42,324,2664,22896,203256,1849392,17156448,161663040,1543053888,
%T 14887836288,144963737856,1422685140480,14058304458624,
%U 139754913276672,1396721001457152,14025182471414784,141432971217841152,1431708373864249344,14543342842406252544,148198801896234491904
%N Expansion of (1-6*x-sqrt((1-6*x)^2-4*6*x^2))/(2*6*x^2).
%C Series reversion of x/(1+6x+6x^2). Transform of 6^n under the matrix A107131. A row of A107267.
%C 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
%H Vincenzo Librandi, <a href="/A107266/b107266.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-6*x-sqrt(1-12*x+12*x^2))/(12*x^2).
%F a(n) = Sum_{k=0..n} 1/(k+1) * C(k+1,n-k+1) * C(n,k) * 6^k.
%F 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
%F 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
%F a(n) ~ sqrt(44+18*sqrt(6))*(6+2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%F 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
%t CoefficientList[Series[(1-6*x-Sqrt[1-12*x+12*x^2])/(12*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (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
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 15 2005
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