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A027081 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027052.

%I A027081 #12 Nov 07 2019 21:11:38
%S A027081 8,56,438,3574,29738,249200,2094902,17648718,148968822,1259807224,
%T A027081 10674450652,90618393250,770728674864,6567151658496,56054864624310,
%U A027081 479267092351534,4104271159315190,35200977081482376,302343415930398696,2600408469332918538,22394817457275426524
%N A027081 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027052.
%H A027081 G. C. Greubel, <a href="/A027081/b027081.txt">Table of n, a(n) for n = 3..1000</a>
%p A027081 T:= proc(n, k) option remember;
%p A027081       if k<0 or k>2*n then 0
%p A027081     elif k=0 or k=2 or k=2*n then 1
%p A027081     elif k=1 then 0
%p A027081     else add(T(n-1, k-j), j=1..3)
%p A027081       fi
%p A027081     end:
%p A027081 seq( add(T(n,k)*T(n,k+3), k=0..2*n-3), n=3..30); # _G. C. Greubel_, Nov 07 2019
%t A027081 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, k+3], {k, 0, 2*n-3}], {n,3,30}] (* _G. C. Greubel_, Nov 07 2019 *)
%o A027081 (Sage)
%o A027081 @CachedFunction
%o A027081 def T(n, k):
%o A027081     if (k<0 or k>2*n): return 0
%o A027081     elif (k==0 or k==2 or k==2*n): return 1
%o A027081     elif (k==1): return 0
%o A027081     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027081 [sum(T(n,k)*T(n,k+3) for k in (0..2*n-3)) for n in (3..30)] # _G. C. Greubel_, Nov 07 2019
%K A027081 nonn
%O A027081 3,1
%A A027081 _Clark Kimberling_
%E A027081 More terms from _Sean A. Irvine_, Oct 22 2019