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a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026769.
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%I #12 Mar 19 2020 14:09:47

%S 1,3,7,16,35,78,171,383,849,1919,4301,9807,22193,50993,116349,269094,

%T 618277,1437916,3323277,7764996,18035275,42304904,98668223,232198092,

%U 543453693,1282401208,3010275001,7119589730,16753996391

%N a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026769.

%H G. C. Greubel, <a href="/A026778/b026778.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 n=2 and k=1 then 2;

%p elif 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(add(T(j,k), k=0..n), j=0..n), n=0..30); # _G. C. Greubel_, Nov 01 2019

%t 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[j,k], {k,0,n}, {j,0,n}], {n,0,30}] (* _G. C. Greubel_, Nov 01 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if n < 0:

%o return 0

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

%o elif (n==2 and k==1): return 2

%o elif (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(sum(T(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 01 2019

%Y Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026779.

%K nonn

%O 0,2

%A _Clark Kimberling_