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%I #9 Jun 16 2024 02:11:52
%S 1,1,4,6,18,27,72,111,283,447,1112,1791,4381,7167,17305,28671,68497,
%T 114687,271560,458751,1077949,1835007,4283069,7340031,17031503,
%U 29360127,67768777,117440511,269797323,469762047,1074583315,1879048191
%N a(n) = Sum_{k=0..floor(n/2)} A026615(n, k).
%H G. C. Greubel, <a href="/A026623/b026623.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = ( 2*(7*n-24)*(7*n-29)*(n-1)*a(n-1) + 4*(7*n-8)*(7*n-31)*(n-3)*a(n-2) - 8*(7*n-15)*(7*n-24)*(n-4)*a(n-3) - (147*n^3 - 1603*n^2 + 5896*n - 7152))/(n*(7*n-31)*(7*n-22)), for n > 3, with a(0) = a(1) = 1, a(2) = 4, and a(3) = 6. - _G. C. Greubel_, Jun 15 2024
%t T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n -1, k-1] + T[n-1,k]]]; (* T = A026615 *)
%t A026623[n_]:= Sum[T[n, k], {k, 0, Floor[n/2]}];
%t Table[A026623[n], {n,0,40}] (* _G. C. Greubel_, Jun 15 2024 *)
%o (Magma)
%o I:=[1,4,6]; [1] cat [n le 3 select I[n] else ( 2*(7*n-24)*(7*n-29)*(n-1)*Self(n-1) + 4*(7*n-8)*(7*n-31)*(n-3)*Self(n-2) - 8*(7*n-15)*(7*n-24)*(n-4)*Self(n-3) - (147*n^3 - 1603*n^2 + 5896*n - 7152))/(n*(7*n-31)*(7*n-22)): n in [1..41]]; // _G. C. Greubel_, Jun 15 2024
%o (SageMath)
%o @CachedFunction
%o def T(n, k): # T = A026615
%o if k==0 or k==n: return 1
%o elif k==1 or k==n-1: return 2*n-1
%o else: return T(n-1, k-1) + T(n-1, k)
%o def A026623(n): return sum(T(n,k) for k in range((n//2)+1))
%o [A026623(n) for n in range(41)] # _G. C. Greubel_, Jun 15 2024
%Y Cf. A026615, A026616, A026617, A026618, A026619, A026620, A026621.
%Y Cf. A026622, A026624, A026625, A026956, A026957, A026958, A026959.
%Y Cf. A026960.
%K nonn
%O 0,3
%A _Clark Kimberling_
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