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

0,2

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.

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A026736, A111279, A059714.

Sequence in context: A150196 A366082 A148491 * A111279 A150197 A150198

Adjacent sequences: A026734 A026735 A026736 * A026738 A026739 A026740

Keyword

nonn

Author

Clark Kimberling

Status

approved