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%I #12 Dec 14 2021 02:44:28
%S 1,1,4,8,23,54,143,354,914,2306,5907,15012,38368,97804,249865,637834,
%T 1629729,4163398,10640753,27196246,69526562,177757762,454541197,
%U 1162403180,2972953385,7604223184,19451741733,49761433640,127308417226
%N Row sums of A026584.
%H G. C. Greubel, <a href="/A026596/b026596.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} A026584(n, k).
%F Conjecture: n*a(n) -3*(n-1)*a(n-1) -(5*n-6)*a(n-2) +3*(5*n-13)*a(n-3) +2*(4*n-9)*a(n-4) -8*(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 a[n_]:=a[n]= Sum[T[n,k], {k,0,n}];
%t Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Dec 13 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 @CachedFunction
%o def A026596(n): return sum( T(n, j) for j in (0..n) )
%o [A026596(n) for n in (0..40)] # _G. C. Greubel_, Dec 13 2021
%Y Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.
%K nonn
%O 0,3
%A _Clark Kimberling_
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