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A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.

%I A026598 #11 Dec 15 2021 01:18:49
%S A026598 1,2,6,14,37,91,234,588,1502,3808,9715,24727,63095,160899,410764,
%T A026598 1048598,2678327,6841725,17482478,44678724,114205286,291963048,
%U A026598 746504245,1908907425,4881860810,12486083994,31937825727,81699259367
%N A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584.
%H A026598 G. C. Greubel, <a href="/A026598/b026598.txt">Table of n, a(n) for n = 0..500</a>
%F A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} A026584(i, j).
%F A026598 Conjecture: n*a(n) - (4*n-3)*a(n-1) - (2*n-3)*a(n-2) + 5*(4*n-9)*a(n-3) - 7*(n-3)*a(n-4) - 6*(4*n-15)*a(n-5) + 8*(2*n-9)*a(n-6) = 0. - _R. J. Mathar_, Jun 23 2013
%t A026598 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, 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 A026598 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[i,j], {i,0,n}, {j,0,i}]];
%t A026598 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Dec 15 2021 *)
%o A026598 (Sage)
%o A026598 @CachedFunction
%o A026598 def T(n, k):  # T = A026584
%o A026598     if (k==0 or k==2*n): return 1
%o A026598     elif (k==1 or k==2*n-1): return (n//2)
%o A026598     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 A026598 @CachedFunction
%o A026598 def A026598(n): return sum(sum(T(i,j) for j in (0..i)) for i in (0..n))
%o A026598 [A026598(n) for n in (0..40)] # _G. C. Greubel_, Dec 15 2021
%Y A026598 Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026599, A027282, A027283, A027284, A027285, A027286.
%K A026598 nonn
%O A026598 0,2
%A A026598 _Clark Kimberling_