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A027070 a(n) = diagonal sum of right-justified array T given by A027052.

%I A027070 #12 Nov 07 2019 08:27:42
%S A027070 1,1,1,4,6,12,31,73,183,476,1248,3322,8943,24271,66355,182538,504824,
%T A027070 1402682,3913585,10959499,30792445,86775340,245204312,694603032,
%U A027070 1972115945,5610955925,15994866669,45677496204,130661330526,374339736820,1074025873959,3085699969569,8876601230175
%N A027070 a(n) = diagonal sum of right-justified array T given by A027052.
%H A027070 G. C. Greubel, <a href="/A027070/b027070.txt">Table of n, a(n) for n = 0..750</a>
%F A027070 a(n) = Sum_{k=0..n} A027052(n-k, 2*n-3*k). - _Sean A. Irvine_, Oct 22 2019
%p A027070 T:= proc(n, k) option remember;
%p A027070       if k<0 or k>2*n then 0
%p A027070     elif k=0 or k=2 or k=2*n then 1
%p A027070     elif k=1 then 0
%p A027070     else add(T(n-1, k-j), j=1..3)
%p A027070       fi
%p A027070     end:
%p A027070 seq( add(T(n-k,2*n-3*k), k=0..n), n=0..35); # _G. C. Greubel_, Nov 06 2019
%t A027070 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[T[n-k, 2*n-3*k], {k, 0, n}], {n,0,35}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027070 (Sage)
%o A027070 @CachedFunction
%o A027070 def T(n, k):
%o A027070     if (k<0 or k>2*n): return 0
%o A027070     elif (k==0 or k==2 or k==2*n): return 1
%o A027070     elif (k==1): return 0
%o A027070     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027070 [sum(T(n-k,2*n-3*k) for k in (0..n)) for n in (0..35)] # _G. C. Greubel_, Nov 06 2019
%K A027070 nonn
%O A027070 0,4
%A A027070 _Clark Kimberling_
%E A027070 More terms from _Sean A. Irvine_, Oct 22 2019