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%I #13 Dec 13 2021 03:05:39
%S 1,1,5,9,28,62,167,399,1024,2518,6359,15819,39759,99427,249699,626203,
%T 1573524,3953446,9943905,25019005,62994733,158680545,399936573,
%U 1008438757,2543992514,6420413940,16210331727,40943722115,103453402718
%N a(n) = T(n, n-2), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=2.
%H G. C. Greubel, <a href="/A026587/b026587.txt">Table of n, a(n) for n = 2..1000</a>
%F a(n) = A026584(n, n-2).
%F Conjecture: (n+2)*a(n) = (3*n+2)*a(n-1) +(3*n+2)*a(n-2) -(11*n-16)*a(n-3) -2*(n-3)*a(n-4) +4*(2*n-9)*a(n-5). - _R. J. Mathar_, Jun 23 2013
%t T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n+k], T[n-1, k-2] + T[n-1, k], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k] ]]]; (* T = A026584 *)
%t Table[T[n, n-2], {n, 2, 40}] (* _G. C. Greubel_, Dec 12 2021 *)
%o (Sage)
%o @CachedFunction
%o def T(n, k): # T = A026584
%o if (k==0 or k==2*n): return 1
%o elif (k==1 or k==2*n-1): return (n//2)
%o else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o [T(n, n-2) for n in (2..40)] # _G. C. Greubel_, Dec 12 2021
%Y Cf. A026584, A026585, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.
%K nonn
%O 2,3
%A _Clark Kimberling_
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