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a(n) = (1/2) * A027052(n, 2n-1).
2

%I #11 Nov 06 2019 04:26:42

%S 1,2,4,9,21,51,128,329,861,2285,6132,16606,45313,124446,343680,953753,

%T 2658133,7436541,20875972,58783892,165989825,469903672,1333359488,

%U 3791535934,10802911297,30836181436,88169413364,252500533673,724182805389,2079862921763,5981150599872

%N a(n) = (1/2) * A027052(n, 2n-1).

%H G. C. Greubel, <a href="/A027057/b027057.txt">Table of n, a(n) for n = 2..750</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( T(n,2*n-1)/2, n=2..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[T[n,2*n-1]/2, {n,2,30}] (* _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 [T(n,2*n-1)/2 for n in (2..30)] # _G. C. Greubel_, Nov 06 2019

%K nonn

%O 2,2

%A _Clark Kimberling_

%E More terms from _Sean A. Irvine_, Oct 22 2019