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Expansion of (1-x-sqrt(1-2x-19x^2))/(10x^2).
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%I #28 Dec 12 2022 06:09:58

%S 1,1,6,16,81,301,1451,6231,29891,137731,666976,3193026,15658831,

%T 76719891,380788006,1894818776,9502977851,47822585931,241944876266,

%U 1228151169656,6258922649451,31992657321551,164040821525031

%N Expansion of (1-x-sqrt(1-2x-19x^2))/(10x^2).

%C Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 5 colors (i.e. Motzkin paths with the up steps in 5 colors). Series reversion of x/(1+x+5x^2). - _Paul Barry_, May 16 2005

%H Vincenzo Librandi, <a href="/A091148/b091148.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A014434(n+1)/5.

%F G.f.: 2/(1-x+sqrt(1-2x-19x^2)).

%F a(n) = sum{k=0..n, binomial(n, k)5^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A000108(n).

%F a(n) = sum{k=0..n, C(n, 2k)C(k)5^k}; - _Paul Barry_, May 16 2005

%F D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +19*(1-n)*a(n-2)=0. - _R. J. Mathar_, Sep 26 2012

%F a(n) ~ 1/10*sqrt(230+61*sqrt(5))/(n^(3/2)*sqrt(Pi))*(1+2*sqrt(5))^n. - _Vaclav Kotesovec_, Sep 29 2012

%F G.f.: 1/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - ....))))), a continued fraction. - _Ilya Gutkovskiy_, May 26 2017

%t CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 19 x^2]) / (10 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, May 10 2013 *)

%o (PARI) x='x+O('x^66); Vec((1-x-sqrt(1-2*x-19*x^2))/(10*x^2)) \\ _Joerg Arndt_, May 11 2013

%Y Cf. A217275.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Dec 22 2003