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a(n) = T(2n+1,n+2), T given by A026736.
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%I #27 Nov 22 2024 07:16:33

%S 1,5,21,85,342,1380,5598,22836,93640,385734,1595232,6619374,27545269,

%T 114901685,480282369,2011058681,8433331523,35410037683,148842787215,

%U 626234799703,2636930617597,11111302351505,46848507630321,197631791675365

%N a(n) = T(2n+1,n+2), T given by A026736.

%H Robert Israel, <a href="/A026855/b026855.txt">Table of n, a(n) for n = 1..1580</a>

%F G.f.: (x*C(x)^4)/(1 - x/sqrt(1 - 4*x)) where C(x) is the g.f. for Catalan numbers A000108. - _David Callan_, Jan 16 2016

%F a(n) ~ (3 - sqrt(5))^4 * (2 + sqrt(5))^(n+2) / (16*sqrt(5)). - _Vaclav Kotesovec_, Jul 18 2019

%F D-finite with recurrence -2*(n+3)*(26*n-53)*a(n) +(627*n^2-491*n-2440)*a(n-1) +2*(-1234*n^2+3198*n+337)*a(n-2) +(2957*n^2-13637*n+15888)*a(n-3) +2*(211*n-368)*(2*n-5)*a(n-4)=0. - _R. J. Mathar_, Nov 22 2024

%p gf := ((-2*x^3+12*x^2-7*x+1)*sqrt(1-4*x)+16*x^3-24*x^2+9*x-1)/(2*(x^2+4*x-1)*x^3):

%p S:= series(gf,x,40):

%p seq(coeff(S,x,j),j=1..30); # _Robert Israel_, Jan 17 2016

%t CoefficientList[Series[(1-Sqrt[1-4x])^4/(16*x^4*(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* _David Callan_, Jan 16 2016 *)

%o (PARI) my(x='x+O('x^30)); Vec( sqrt(1-4*x)*(1-sqrt(1-4*x))^4/(16*x^3*(sqrt(1-4*x) -x)) ) \\ _G. C. Greubel_, Jul 17 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^4/(16*x^3*(Sqrt(1-4*x) -x)) )); // _G. C. Greubel_, Jul 17 2019

%o (Sage) a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^4/(16*x^3*(sqrt(1-4*x) -x))).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Jul 17 2019

%Y Cf. A000108, A026736.

%K nonn,changed

%O 1,2

%A _Clark Kimberling_