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a(n) = T(n, floor(n/2)), T given by A026747.
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%I #9 Oct 29 2019 21:10:10

%S 1,1,3,4,11,17,44,74,184,327,790,1461,3452,6584,15278,29879,68290,

%T 136391,307696,625731,1395696,2883357,6367199,13338421,29193025,

%U 61920497,134442102,288368511,621609060,1346873365,2884432810

%N a(n) = T(n, floor(n/2)), T given by A026747.

%H G. C. Greubel, <a href="/A026753/b026753.txt">Table of n, a(n) for n = 0..1000</a>

%p A026747 := proc(n,k) option remember;

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

%p elif type(n,'even') and k <= n/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(A026747(n,floor(n/2)), n=0..30); # _G. C. Greubel_, Oct 29 2019

%t T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[EvenQ[n] && k<=n/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[T[n, Floor[n/2]], {n,0,30}] (* _G. C. Greubel_, Oct 29 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

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

%o elif (mod(n,2)==0 and k<=n/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 [T(n, floor(n/2)) for n in (0..30)] # _G. C. Greubel_, Oct 29 2019

%Y Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026754, A026755, A026756, A026757.

%K nonn

%O 0,3

%A _Clark Kimberling_