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

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, 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, 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

Offset: 0

Author

Paul Barry, Sep 17 2004

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins as:
   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, 14,  7,  1;
  -1, 0, 0, 0, 2, 16, 20,  8, 1;
   1, 0, 0, 0, 0,  9, 30, 27, 9, 1;
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1325

Crossrefs

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

Row sums are A098600.

Diagonal sums are A098601.

Apart from signs, same as A100218.

Very similar to triangle A111125.

Programs

Magma

A098599:= func< n,k | n eq 0 select 1 else Binomial(k, n-k) + Binomial(k-1, n-k-1) >;
[A098599(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2024

Mathematica

T[n_, k_]:= If[n==0, 1, Binomial[k,n-k] +Binomial[k-1,n-k-1]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 27 2024 *)

SageMath

def A098599(n,k): return 1 if n==0 else binomial(k, n-k) + binomial(k-1, n-k-1)
flatten([[A098599(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 27 2024

Formula

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

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

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

From G. C. Greubel, Mar 27 2024: (Start)

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

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

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

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

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