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%I #10 Dec 13 2021 02:26:25
%S 1,2,9,22,69,178,497,1294,3452,8964,23430,60556,156663,403214,1037191,
%T 2660978,6821200,17459732,44657246,114117628,291449047,743904326,
%U 1897956899,4840429962,12340947855,31455453822,80158533099
%N a(n) = T(n,n-4), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.
%H G. C. Greubel, <a href="/A026589/b026589.txt">Table of n, a(n) for n = 4..1000</a>
%F a(n) = A026584(n, n-4).
%F Conjecture: -(n+4)*(65*n-269)*a(n) +(-65*n^2+140*n+1933)*a(n-1) +(809*n^2-2431*n-4514)*a(n-2) +(-123*n^2+2496*n-205)*a(n-3) +2*(-726*n^2+3737*n-4395)*a(n-4) +8*(56*n-215)*(2*n-9)*a(n-5) = 0. - _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-4], {n, 4, 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-4) for n in (4..40)] # _G. C. Greubel_, Dec 12 2021
%Y Cf. A026584, A026585, A026587, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.
%K nonn
%O 4,2
%A _Clark Kimberling_
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