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A027068 a(n) = Sum_{i=0..n} Sum_{j=i..2*i} A027052(i, j).

%I A027068 #17 Nov 07 2019 08:28:07
%S A027068 1,2,6,16,43,120,340,972,2793,8050,23256,67324,195275,567448,1651830,
%T A027068 4816328,14064569,41128626,120425604,353022920,1035983443,3043189688,
%U A027068 8947381566,26328236756,77531471737,228475334594,673725464150
%N A027068 a(n) = Sum_{i=0..n} Sum_{j=i..2*i} A027052(i, j).
%H A027068 G. C. Greubel, <a href="/A027068/b027068.txt">Table of n, a(n) for n = 0..1000</a>
%p A027068 T:= proc(n, k) option remember;
%p A027068       if k<0 or k>2*n then 0
%p A027068     elif k=0 or k=2 or k=2*n then 1
%p A027068     elif k=1 then 0
%p A027068     else add(T(n-1, k-j), j=1..3)
%p A027068       fi
%p A027068     end:
%p A027068 seq( add(add(T(k,j), j=k..2*k), k=0..n), n=0..30); # _G. C. Greubel_, Nov 06 2019
%t A027068 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j,3} ]]]]; Table[Sum[Sum[T[i, j], {j, i, 2*i}], {i, 0, n}], {n,0,30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027068 (Sage)
%o A027068 @CachedFunction
%o A027068 def T(n, k):
%o A027068     if (k<0 or k>2*n): return 0
%o A027068     elif (k==0 or k==2 or k==2*n): return 1
%o A027068     elif (k==1): return 0
%o A027068     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027068 [sum(sum(T(k,j) for j in (k..2*k)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 06 2019
%Y A027068 Cf. A027052. Partial sums of A027067.
%K A027068 nonn
%O A027068 0,2
%A A027068 _Clark Kimberling_
%E A027068 Title corrected by _Sean A. Irvine_, Oct 22 2019