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A026598 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026584. 17

%I #11 Dec 15 2021 01:18:49

%S 1,2,6,14,37,91,234,588,1502,3808,9715,24727,63095,160899,410764,

%T 1048598,2678327,6841725,17482478,44678724,114205286,291963048,

%U 746504245,1908907425,4881860810,12486083994,31937825727,81699259367

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

%H G. C. Greubel, <a href="/A026598/b026598.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{i=0..n} Sum_{j=0..i} A026584(i, j).

%F 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 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 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[i,j], {i,0,n}, {j,0,i}]];

%t Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Dec 15 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 A026598(n): return sum(sum(T(i,j) for j in (0..i)) for i in (0..n))

%o [A026598(n) for n in (0..40)] # _G. C. Greubel_, Dec 15 2021

%Y Cf. A026584, A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026599, A027282, A027283, A027284, A027285, A027286.

%K nonn

%O 0,2

%A _Clark Kimberling_

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)