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A027072 a(n) = self-convolution of row n of array T given by A027052.

%I A027072 #10 Nov 07 2019 08:28:53
%S A027072 1,2,3,12,53,222,899,3540,13657,51882,194727,723760,2668453,9771870,
%T A027072 35577935,128887616,464885073,1670362418,5981289455,21352860808,
%U A027072 76020123293,269977176422,956644165503,3382864303648,11940005836537
%N A027072 a(n) = self-convolution of row n of array T given by A027052.
%H A027072 G. C. Greubel, <a href="/A027072/b027072.txt">Table of n, a(n) for n = 0..1000</a>
%F A027072 a(n) = Sum_{k=0..2*n} T(n,k)*T(n,2*n-k), where T = A027052. - _G. C. Greubel_, Nov 06 2019
%p A027072 T:= proc(n, k) option remember;
%p A027072       if k<0 or k>2*n then 0
%p A027072     elif k=0 or k=2 or k=2*n then 1
%p A027072     elif k=1 then 0
%p A027072     else add(T(n-1, k-j), j=1..3)
%p A027072       fi
%p A027072     end:
%p A027072 seq( add(T(n,k)*T(n,2*n-k), k=0..2*n), n=0..30); # _G. C. Greubel_, Nov 06 2019
%t A027072 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]*T[n,2*n-k], {k,0,2*n}], {n,0,30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027072 (Sage)
%o A027072 @CachedFunction
%o A027072 def T(n, k):
%o A027072     if (k<0 or k>2*n): return 0
%o A027072     elif (k==0 or k==2 or k==2*n): return 1
%o A027072     elif (k==1): return 0
%o A027072     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027072 [sum(T(n,k)*T(n,2*n-k) for k in (0..2*n)) for n in (0..30)] # _G. C. Greubel_, Nov 06 2019
%K A027072 nonn
%O A027072 0,2
%A A027072 _Clark Kimberling_