cp's OEIS Frontend

A026737 a(n) = T(2*n,n), T given by A026736.

%I A026737 #44 Sep 08 2022 08:44:49
%S A026737 1,2,6,21,79,309,1237,5026,20626,85242,354080,1476368,6173634,
%T A026737 25873744,108628550,456710589,1922354351,8098984433,34147706833,
%U A026737 144068881455,608151037123,2568318694867,10850577045131,45856273670841
%N A026737 a(n) = T(2*n,n), T given by A026736.
%C A026737 Is this the same sequence as A111279? - _Andrew S. Plewe_, May 09 2007
%C A026737 Yes, see the Callan reference "A bijection...". - _Joerg Arndt_, Feb 29 2016
%H A026737 G. C. Greubel, <a href="/A026737/b026737.txt">Table of n, a(n) for n = 0..1000</a>
%H A026737 David Callan, <a href="http://arxiv.org/abs/1602.08347">A bijection for two sequences in OEIS</a>, arXiv:1602.08347 [math.CO], 2016.
%F A026737 G.f.: (1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2)). - _G. C. Greubel_, Jul 16 2019
%F A026737 a(n) ~ (47 - 21*sqrt(5)) * (2 + sqrt(5))^(n+2) / (2*sqrt(5)). - _Vaclav Kotesovec_, Jul 18 2019
%F A026737 G.f. G satisfies 0 = G^2*(x^3 + 4*x^2 - x) + G*(4*x^2 - 5*x + 1) + 4*x - 1. - _F. Chapoton_, Oct 16 2021
%t A026737 T[_, 0]=T[n_, n_]=1; T[n_, k_]:= T[n, k]= If[EvenQ[n] && k==(n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]];
%t A026737 a[n_] := T[2n, n];
%t A026737 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 22 2018 *)
%t A026737 CoefficientList[Series[(1-5x+4x^2 -(1-5x)*Sqrt[1-4x])/(2*x*(1-4x-x^2)), {x, 0, 30}], x] (* _G. C. Greubel_, Jul 16 2019 *)
%o A026737 (PARI) my(x='x+O('x^30)); Vec((1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2))) \\ _G. C. Greubel_, Jul 16 2019
%o A026737 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-5*x+4*x^2 -(1-5*x)*Sqrt(1-4*x))/(2*x*(1-4*x-x^2)) )); // _G. C. Greubel_, Jul 16 2019
%o A026737 (Sage)
%o A026737 @CachedFunction
%o A026737 def T(n, k):
%o A026737     if (k==0 or k==n): return 1
%o A026737     elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o A026737     else: return T(n-1, k-1) + T(n-1, k)
%o A026737 [T(2*n, n) for n in (0..30)] # _G. C. Greubel_, Jul 16 2019
%Y A026737 Cf. A026736, A111279, A059714.
%K A026737 nonn
%O A026737 0,2
%A A026737 _Clark Kimberling_