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

1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Is this the same sequence as A111279? - Andrew S. Plewe, May 09 2007

Yes, see the Callan reference "A bijection...". - Joerg Arndt, Feb 29 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• David Callan, A bijection for two sequences in OEIS, arXiv:1602.08347 [math.CO], 2016.

Crossrefs

Cf. A026736, A111279, A059714.

Programs

Magma

R:=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

Mathematica

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]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 22 2018 *)
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 *)

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

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    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)
    else: return T(n-1, k-1) + T(n-1, k)
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Jul 16 2019

Formula

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

a(n) ~ (47 - 21*sqrt(5)) * (2 + sqrt(5))^(n+2) / (2*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

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