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A026777 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026769.

%I A026777 #7 Nov 01 2019 22:23:28
%S A026777 1,1,3,5,14,26,70,138,362,742,1912,4028,10249,22033,55547,121273,
%T A026777 303641,670997,1671233,3729071,9250099,20803231,51437219,116436313,
%U A026777 287152067,653567143,1608416195,3677760541,9035150126,20741496354
%N A026777 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026769.
%H A026777 G. C. Greubel, <a href="/A026777/b026777.txt">Table of n, a(n) for n = 0..1000</a>
%p A026777 T:= proc(n,k) option remember;
%p A026777    if n<0 then 0;
%p A026777    elif k=0 or k=n then 1;
%p A026777    elif n=2 and k=1 then 2;
%p A026777    elif k <= (n-1)/2 then
%p A026777         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026777    else
%p A026777        procname(n-1,k-1)+procname(n-1,k) ;
%p A026777    end if ;
%p A026777 end proc;
%p A026777 seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Nov 01 2019
%t A026777 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[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_, Nov 01 2019 *)
%o A026777 (Sage)
%o A026777 @CachedFunction
%o A026777 def T(n, k):
%o A026777     if (k==0 or k==n): return 1
%o A026777     elif (n==2 and k==1): return 2
%o A026777     elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026777     else: return T(n-1,k-1) + T(n-1,k)
%o A026777 [sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Nov 01 2019
%Y A026777 Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026778, A026779.
%K A026777 nonn
%O A026777 0,3
%A A026777 _Clark Kimberling_