A026765 a(n) = Sum_{k=0..n} T(n,k), T given by A026758.

1, 2, 4, 9, 18, 41, 82, 188, 376, 867, 1734, 4020, 8040, 18735, 37470, 87735, 175470, 412715, 825430, 1949624, 3899248, 9245721, 18491442, 44003717, 88007434, 210121733, 420243466, 1006390014, 2012780028, 4833517551

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026766, A026767, A026768.

Programs

Maple

T:= proc(n,k) option remember;
   if n<0 then 0;
   elif k=0 or k = n then 1;
   elif type(n,'odd') and k <= (n-1)/2 then
        procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
   else
       procname(n-1,k-1)+procname(n-1,k) ;
   end if ;
end proc;
seq(add(T(n,k), k=0..n), n=0..30); # G. C. Greubel, Oct 31 2019

Mathematica

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k<=(n - 1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]; Table[Sum[T[n,k],{k,0,n}], {n, 0, 30}] (* G. C. Greubel, Oct 31 2019 *)

Sage

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k<=(n-1)/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)
[sum(T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 31 2019

Formula

Conjecture: G.f.: -(1-2*x-5*x^2+10*x^3 - sqrt(1-10*x^2+29*x^4-20*x^6) )/(2*x*(1-2*x-5*x^2+10*x^3)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Conjecture: (n+1)*a(n) -2*a(n-1) +2*(-5*n+3)*a(n-2) +12*a(n-3) +(29*n-71)*a(n-4) -10*a(n-5) +20*(-n+5)*a(n-6)=0. - R. J. Mathar, Jun 30 2013

Conjecture: a(n) ~ (2+sqrt(5) + (-1)^n*(2-sqrt(5))) * 5^(n/2) / sqrt(2*Pi*n). - Vaclav Kotesovec, Feb 12 2014