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A026732 a(n) = Sum_{k=0..n} T(n,k), T given by A026725.

%I A026732 #9 Oct 26 2019 15:38:52
%S A026732 1,2,4,9,18,40,80,176,352,769,1538,3343,6686,14477,28954,62505,125010,
%T A026732 269216,538432,1157244,2314488,4966260,9932520,21282622,42565244,
%U A026732 91096110,182192220,389515284,779030568,1664015246,3328030492
%N A026732 a(n) = Sum_{k=0..n} T(n,k), T given by A026725.
%H A026732 G. C. Greubel, <a href="/A026732/b026732.txt">Table of n, a(n) for n = 0..1000</a>
%F A026732 Conjecture: +(-n+1)*a(n) +2*a(n-1) +3*(3*n-7)*a(n-2) -10*a(n-3) +(-23*n+95)*a(n-4) +6*a(n-5) +(11*n-95)*a(n-6) +2*a(n-7) +4*(n-7)*a(n-8)=0. - _R. J. Mathar_, Oct 26 2019
%p A026732 A026732 := proc(n)
%p A026732     add(A026725(n,k),k=0..n) ;
%p A026732 end proc:
%p A026732 seq(A026732(n),n=0..10) ; # _R. J. Mathar_, Oct 26 2019
%t A026732 T[n_, k_]:= T[n, k]= 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 26 2019 *)
%o A026732 (PARI) T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
%o A026732 vector(31, n, sum(k=0,n-1, T(n-1,k)) ) \\ _G. C. Greubel_, Oct 26 2019
%o A026732 (Sage)
%o A026732 @CachedFunction
%o A026732 def T(n, k):
%o A026732     if (k==0 or k==n): return 1
%o A026732     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 A026732     else: return T(n-1, k-1) + T(n-1, k)
%o A026732 [sum(T(n, k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Oct 26 2019
%o A026732 (GAP)
%o A026732 T:= function(n,k)
%o A026732     if k=0 or k=n then return 1;
%o A026732     elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o A026732     else return T(n-1, k-1) + T(n-1, k);
%o A026732     fi;
%o A026732   end;
%o A026732 List([0..30], n-> Sum([0..n], k-> T(n,k) )); # _G. C. Greubel_, Oct 26 2019
%K A026732 nonn
%O A026732 0,2
%A A026732 _Clark Kimberling_