VOOZH about

URL: https://oeis.org/A098599

⇱ A098599 - OEIS

login
A098599
Riordan array ((1+2*x)/(1+x), (1+x)).
7
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
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
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
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Sep 17 2004
STATUS
approved