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

0,2

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A026732 A171003 A094291 * A264649 A259803 A032175

Adjacent sequences: A026762 A026763 A026764 * A026766 A026767 A026768

Keyword

nonn

Author

Clark Kimberling

Status

approved