A098599 - OEIS

%I #17 Mar 27 2024 08:59:58

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

%H G. C. Greubel, <a href="/A098599/b098599.txt">Table of n, a(n) for n = 0..1325</a>

%F Triangle: T(n, k) = binomial(k, n-k) + binomial(k-1, n-k-1), with T(0, 0) = 1.

%F Sum_{k=0..n} T(n, k) = A098600(n) (row sums).

%F 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 From _G. C. Greubel_, Mar 27 2024: (Start)

%F T(2*n, n) = A040000(n).

%F T(2*n+1, n) = A000007(n).

%F T(2*n-1, n) = A005408(n-1), n >= 1.

%F Sum_{k=0..n} (-1)^k*T(n, k) = A079757(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A098601(n). (End)

%e Triangle begins as:

%e    1;

%e    1, 1;

%e   -1, 2, 1;

%e    1, 0, 3, 1;

%e   -1, 0, 2, 4, 1;

%e    1, 0, 0, 5, 5,  1;

%e   -1, 0, 0, 2, 9,  6,  1;

%e    1, 0, 0, 0, 7, 14,  7,  1;

%e   -1, 0, 0, 0, 2, 16, 20,  8, 1;

%e    1, 0, 0, 0, 0,  9, 30, 27, 9, 1;

%t T[n_, k_]:= If[n==0, 1, Binomial[k,n-k] +Binomial[k-1,n-k-1]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 27 2024 *)

%o (Magma)

%o A098599:= func< n,k | n eq 0 select 1 else Binomial(k, n-k) + Binomial(k-1, n-k-1) >;

%o [A098599(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 27 2024

%o (SageMath)

%o def A098599(n,k): return 1 if n==0 else binomial(k, n-k) + binomial(k-1, n-k-1)

%o flatten([[A098599(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 27 2024

%Y Cf. A000007, A005408, A029635, A040000, A079757, A098601.

%Y Row sums are A098600.

%Y Diagonal sums are A098601.

%Y Apart from signs, same as A100218.

%Y Very similar to triangle A111125.

%K easy,sign,tabl

%O 0,5

%A _Paul Barry_, Sep 17 2004