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A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

%I A098599 #17 Mar 27 2024 08:59:58
%S A098599 1,1,1,-1,2,1,1,0,3,1,-1,0,2,4,1,1,0,0,5,5,1,-1,0,0,2,9,6,1,1,0,0,0,7,
%T A098599 14,7,1,-1,0,0,0,2,16,20,8,1,1,0,0,0,0,9,30,27,9,1,-1,0,0,0,0,2,25,50,
%U A098599 35,10,1,1,0,0,0,0,0,11,55,77,44,11,1,-1,0,0,0,0,0,2,36,105,112,54,12,1,1,0,0,0,0,0,0,13,91,182,156,65,13,1
%N A098599 Riordan array ((1+2*x)/(1+x), (1+x)).
%H A098599 G. C. Greubel, <a href="/A098599/b098599.txt">Table of n, a(n) for n = 0..1325</a>
%F A098599 Triangle: T(n, k) = binomial(k, n-k) + binomial(k-1, n-k-1), with T(0, 0) = 1.
%F A098599 Sum_{k=0..n} T(n, k) = A098600(n) (row sums).
%F A098599 T(n,k) = T(n-1,k-1) - T(n-1,k) + 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=1, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Jan 09 2014
%F A098599 From _G. C. Greubel_, Mar 27 2024: (Start)
%F A098599 T(2*n, n) = A040000(n).
%F A098599 T(2*n+1, n) = A000007(n).
%F A098599 T(2*n-1, n) = A005408(n-1), n >= 1.
%F A098599 Sum_{k=0..n} (-1)^k*T(n, k) = A079757(n).
%F A098599 Sum_{k=0..floor(n/2)} T(n-k, k) = A098601(n). (End)
%e A098599 Triangle begins as:
%e A098599    1;
%e A098599    1, 1;
%e A098599   -1, 2, 1;
%e A098599    1, 0, 3, 1;
%e A098599   -1, 0, 2, 4, 1;
%e A098599    1, 0, 0, 5, 5,  1;
%e A098599   -1, 0, 0, 2, 9,  6,  1;
%e A098599    1, 0, 0, 0, 7, 14,  7,  1;
%e A098599   -1, 0, 0, 0, 2, 16, 20,  8, 1;
%e A098599    1, 0, 0, 0, 0,  9, 30, 27, 9, 1;
%t A098599 T[n_, k_]:= If[n==0, 1, Binomial[k,n-k] +Binomial[k-1,n-k-1]];
%t A098599 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 27 2024 *)
%o A098599 (Magma)
%o A098599 A098599:= func< n,k | n eq 0 select 1 else Binomial(k, n-k) + Binomial(k-1, n-k-1) >;
%o A098599 [A098599(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 27 2024
%o A098599 (SageMath)
%o A098599 def A098599(n,k): return 1 if n==0 else binomial(k, n-k) + binomial(k-1, n-k-1)
%o A098599 flatten([[A098599(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 27 2024
%Y A098599 Cf. A000007, A005408, A029635, A040000, A079757, A098601.
%Y A098599 Row sums are A098600.
%Y A098599 Diagonal sums are A098601.
%Y A098599 Apart from signs, same as A100218.
%Y A098599 Very similar to triangle A111125.
%K A098599 easy,sign,tabl
%O A098599 0,5
%A A098599 _Paul Barry_, Sep 17 2004