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%I #21 Jan 30 2020 21:29:15
%S 1,5,30,200,1425,10625,81875,646875,5211875,42659375,353725000,
%T 2965031250,25083859375,213894609375,1836516718750,15863968750000,
%U 137767560546875,1202116083984375,10534061644531250,92664360625000000,817975366904296875,7243402948779296875
%N Expansion of (1-5*x-sqrt((1-5*x)^2-4*5*x^2))/(2*5*x^2).
%C Series reversion of x/(1+5x+5x^2). Transform of 5^n under the matrix A107131. A row of A107267.
%C Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 5 colors and D(1,-1) one color. - _Paul Barry_, May 16 2005
%H Vincenzo Librandi, <a href="/A107265/b107265.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-5*x-sqrt(1-10*x+5*x^2))/(10*x^2).
%F a(n) = sum{k=0..n, (1/(k+1)) * C(k+1,n-k+1) * C(n, k) * 5^k}.
%F E.g.f.: a(n) = n!* [x^n] exp(5*x)*BesselI(1,2*sqrt(5)*x) /(sqrt(5)*x). -_Peter Luschny_, Aug 25 2012
%F D-finite with recurrence: (n+2)*a(n) = 5*(2*n+1)*a(n-1) - 5*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ sqrt(38+17*sqrt(5))*(5+2*sqrt(5))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%F G.f.: 1/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Sep 21 2017
%t CoefficientList[Series[(1-5*x-Sqrt[1-10*x+5*x^2])/(10*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+5*x^2))/(10*x^2)) \\ _Joerg Arndt_, May 15 2013
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 15 2005
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