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A027040 a(n) = self-convolution of row n of array T given by A027023.

%I A027040 #11 Nov 06 2019 22:35:48
%S A027040 1,3,9,31,129,531,2129,8351,32177,122211,458801,1706015,6293169,
%T A027040 23057651,83992313,304424639,1098525761,3948727555,14145206209,
%U A027040 50515602111,179904080257,639103899411,2265253438745,8012421964063
%N A027040 a(n) = self-convolution of row n of array T given by A027023.
%H A027040 G. C. Greubel, <a href="/A027040/b027040.txt">Table of n, a(n) for n = 0..1000</a>
%F A027040 a(n) = Sum_{k=0..2*n} T(n,k)*T(n,2*n-k), where T = A027023. - _G. C. Greubel_, Nov 05 2019
%p A027040 T:= proc(n, k) option remember;
%p A027040       if (n<0 or k>2*n) then 0
%p A027040     elif k<3 or k=2*n then 1
%p A027040     else add(T(n-1, k-j), j=1..3)
%p A027040       fi
%p A027040     end:
%p A027040 seq( add(T(n,k)*T(n,2*n-k), k=0..2*n), n=0..30); # _G. C. Greubel_, Nov 05 2019
%t A027040 T[n_, k_]:= T[n, k]= If[n<0 || k>2*n, 0, If[k<3 || k==2*n, 1, 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 05 2019 *)
%o A027040 (Sage)
%o A027040 @CachedFunction
%o A027040 def T(n, k):
%o A027040     if (n<0 or k>2*n): return 0
%o A027040     elif (k<3 or k==2*n): return 1
%o A027040     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027040 [sum(T(n,k)*T(n,2*n-k) for k in (0..2*n)) for n in (4..30)] # _G. C. Greubel_, Nov 05 2019
%K A027040 nonn
%O A027040 0,2
%A A027040 _Clark Kimberling_