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a(n) = T(2*n, n-2), where T is given by A026552.
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%I #10 Dec 18 2021 01:00:17

%S 1,4,18,74,311,1296,5432,22796,95958,404812,1711600,7250970,30772989,

%T 130810512,556867224,2373764416,10130935783,43285462884,185129287262,

%U 792525473552,3395664830670,14560682746632,62482560679368,268307898599664,1152883194581155,4956738399534376,21323028570642414,91775945084805898

%N a(n) = T(2*n, n-2), where T is given by A026552.

%H G. C. Greubel, <a href="/A026560/b026560.txt">Table of n, a(n) for n = 2..1000</a>

%F a(n) = A026552(2*n, n-2).

%t T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)

%t a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[2*n, n-2]];

%t Table[a[n], {n,2,40}] (* _G. C. Greubel_, Dec 18 2021 *)

%o (Sage)

%o @CachedFunction

%o def T(n,k): # T = A026552

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

%o elif (k==1 or k==2*n-1): return (n+2)//2

%o elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)

%o else: return T(n-1, k) + T(n-1, k-2)

%o [T(2*n,n-2) for n in (2..40)] # _G. C. Greubel_, Dec 18 2021

%Y Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.

%K nonn

%O 2,2

%A _Clark Kimberling_

%E Terms a(20) onward from _G. C. Greubel_, Dec 18 2021