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a(n) = Sum_{k=0..n} T(n,k) * T(n,2n-k), with T given by A027052.
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%I #13 Nov 07 2019 08:28:48

%S 1,1,2,8,31,129,510,1970,7513,28253,105176,388330,1423691,5188577,

%T 18812848,67907520,244160177,874821817,3124747792,11130097846,

%U 39544807851,140180597013,495886522916,1750852227736,6171019594129

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

%H G. C. Greubel, <a href="/A027073/b027073.txt">Table of n, a(n) for n = 0..1000</a>

%p T:= proc(n, k) option remember;

%p if k<0 or k>2*n then 0

%p elif k=0 or k=2 or k=2*n then 1

%p elif k=1 then 0

%p else add(T(n-1, k-j), j=1..3)

%p fi

%p end:

%p seq( add(T(n,k)*T(n,2*n-k), k=0..n), n=0..30); # _G. C. Greubel_, Nov 06 2019

%t 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, n}], {n,0,30}] (* _G. C. Greubel_, Nov 06 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k<0 or k>2*n): return 0

%o elif (k==0 or k==2 or k==2*n): return 1

%o elif (k==1): return 0

%o else: return sum(T(n-1, k-j) for j in (1..3))

%o [sum(T(n,k)*T(n,2*n-k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 06 2019

%K nonn

%O 0,3

%A _Clark Kimberling_