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A027042 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027023.

%I A027042 #13 Nov 05 2019 01:00:44
%S A027042 1,4,11,52,225,920,3695,14464,55593,210776,789995,2933380,10807625,
%T A027042 39556316,143958335,521340016,1879901265,6753038624,24176722555,
%U A027042 86294777316,307179518193,1090771084252,3864614381391,13664531314176,48225146757337,169905685271956,597661852713467
%N A027042 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027023.
%H A027042 G. C. Greubel, <a href="/A027042/b027042.txt">Table of n, a(n) for n = 1..1000</a>
%p A027042 T:= proc(n, k) option remember;
%p A027042       if k<3 or k=2*n then 1
%p A027042     else add(T(n-1, k-j), j=1..3)
%p A027042       fi
%p A027042     end:
%p A027042 seq(add(T(n,k)*T(n,2*n-k), k=0..n-1), n=1..30); # _G. C. Greubel_, Nov 04 2019
%t A027042 T[n_, k_]:= T[n, k]= 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,n-1}], {n,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027042 (Sage)
%o A027042 @CachedFunction
%o A027042 def T(n, k):
%o A027042     if (k<3 or k==2*n): return 1
%o A027042     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027042 [sum(T(n, k)*T(n,2*n-k) for k in (0..n-1)) for n in (1..30)] # _G. C. Greubel_, Nov 04 2019
%K A027042 nonn
%O A027042 1,2
%A A027042 _Clark Kimberling_
%E A027042 More terms from _Sean A. Irvine_, Oct 22 2019