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A026757 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026747.

%I A026757 #10 Oct 29 2019 19:03:08
%S A026757 1,1,2,4,6,11,20,32,58,102,169,302,527,888,1573,2741,4661,8215,14316,
%T A026757 24481,43023,74998,128747,225867,393838,678047,1188201,2072239,
%U A026757 3575728,6261248,10921278,18879372,33040083,57637061
%N A026757 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026747.
%H A026757 G. C. Greubel, <a href="/A026757/b026757.txt">Table of n, a(n) for n = 0..1000</a>
%p A026757 A026747 := proc(n,k) option remember;
%p A026757    if k=0 or k = n then 1;
%p A026757    elif type(n,'even') and k <= n/2 then
%p A026757         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026757    else
%p A026757        procname(n-1,k-1)+procname(n-1,k) ;
%p A026757    end if ;
%p A026757 end proc:
%p A026757 seq(add(A026747(n-k,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Oct 29 2019
%t A026757 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[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Oct 29 2019 *)
%o A026757 (Sage)
%o A026757 @CachedFunction
%o A026757 def T(n, k):
%o A026757     if (k==0 or k==n): return 1
%o A026757     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 A026757     else: return T(n-1,k-1) + T(n-1,k)
%o A026757 [sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Oct 29 2019
%Y A026757 Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026756.
%K A026757 nonn
%O A026757 0,3
%A A026757 _Clark Kimberling_