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a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026758.
10

%I #8 Nov 01 2019 03:54:53

%S 1,1,3,5,13,24,59,115,273,552,1278,2655,6031,12795,28632,61775,136572,

%T 298764,653948,1447225,3141427,7020833,15132512,34106865,73069892,

%U 165903082,353576829,807957495,1714132308,3939206346

%N a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026758.

%H G. C. Greubel, <a href="/A026766/b026766.txt">Table of n, a(n) for n = 0..1000</a>

%p T:= proc(n,k) option remember;

%p if n<0 then 0;

%p elif k=0 or k = n then 1;

%p elif type(n,'odd') and k <= (n-1)/2 then

%p procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;

%p else

%p procname(n-1,k-1)+procname(n-1,k) ;

%p end if ;

%p end proc;

%p seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Oct 31 2019

%t 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,Floor[n/2]}], {n, 0, 30}] (* _G. C. Greubel_, Oct 31 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (n<0): return 0

%o elif (k==0 or k==n): return 1

%o 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)

%o else: return T(n-1,k-1) + T(n-1,k)

%o [sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Oct 31 2019

%Y Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026765, A026767, A026768.

%K nonn

%O 0,3

%A _Clark Kimberling_