%I #31 Jan 30 2020 21:29:15
%S 1,6,37,233,1491,9660,63195,416610,2763595,18426026,123375927,
%T 829053197,5588050069,37764371676,255800207277,1736181639585,
%U 11804962371795,80394249836010,548283258074895,3744067955618403,25596986050620681
%N Expansion of (sqrt(21*x^2 - 10*x + 1) + 7*x - 1) / (2*x*(1 - 7*x)).
%C 3rd binomial transform of C(2n+1,n+1) (A001700); 5th binomial transform of C(n,floor(n/2)) (A001405); 7th binomial transform of (-1)^n*A000108(n) = A168491(n). Hankel transform is (1,1,1,.....). Row sums of Riordan array (1/(1+5x+x^2), x/(1+5x+x^2))^(-1). Counts Motzkin paths with 5 colors for horizontal steps. [Corrected by _Philippe Deléham_, Nov 29 2009]
%C Binomial transform of A005573. 7th binomial transform of A168491. - _Philippe Deléham_, Nov 28 2009
%H Vincenzo Librandi, <a href="/A122898/b122898.txt">Table of n, a(n) for n = 0..200</a>
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F a(n) = Sum{k=0..n, 3^(n-k)C(n,k)C(2k+1,k+1)}.
%F a(n) = Sum{k=0..n, 5^(n-k)C(n,k)C(k,floor(k/2))}.
%F a(n) = Sum{k=0..n, 7^(n-k)C(n,k)*(-1)^k*C(k)} where C(n)=A000108(n).
%F a(n) = Sum{k=0..n, sum{j=0..n, 3^(n-j)*C(n,k)*C(n-k,j-k)*C(j+1,k+1)}}.
%F G.f.: 1/(1-6x-x^2/(1-5x-x^2/(1-5x-x^2/(1-5x-x^2/(1-...(continued fraction). - _Philippe Deléham_, Nov 28 2009
%F D-finite with recurrence: (n+1)*a(n) = 2*(5*n+1)*a(n-1) - 21*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 7^(n+1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 19 2012
%F G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-3*x) - 2*x*(1-3*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-3*x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 24 2013
%F a(n) = (-1)^n*(GegenbauerC(n,-n,5/2) - GegenbauerC(n-1,-n,5/2)). - _Peter Luschny_, May 13 2016
%F E.g.f.: exp(5*x)*(BesselI(0,2*x) + BesselI(1,2*x)). - _Ilya Gutkovskiy_, Sep 20 2017
%p a := n -> (-1)^n*simplify(GegenbauerC(n,-n,5/2) - GegenbauerC(n-1,-n,5/2)):
%p seq(a(n), n=0..21); # _Peter Luschny_, May 13 2016
%t CoefficientList[Series[(Sqrt[21*x^2-10*x+1]+7*x-1)/(2*x*(1-7*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 18 2006