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

0,5

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

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.

Sequence in context: A175417 A136481 A100218 * A326925 A129334 A116399

Adjacent sequences: A098596 A098597 A098598 * A098600 A098601 A098602

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 17 2004

Status

approved