OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1), with T(0, 0) = 1.
T(n, k) = (-1)^floor((n-k+1)/2) * A007318(n, k).
From G. C. Greubel, Dec 02 2022: (Start)
T(2*n, n) = (-1)^binomial(n+1,2) * A000984(n).
T(2*n, n+1) = (-1)^binomial(n,2) * A001791(n), n >= 1.
T(2*n, n-1) = (-1)^binomial(n+2,2) * A001791(n).
T(2*n+1, n-1) = (-1)^binomial(n-1,2) * A002054(n).
T(2*n+1, n+1) = (-1)^binomial(n+1,2) * A001700(n+1).
Sum_{k=0..n} T(n, k) = (-1)^n * A090132(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A108520(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A260192(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A333378(n+1). (End)
MATHEMATICA
A108086[n_, k_]:= (-1)^(Floor[(n-k+1)/2])*Binomial[n, k];
Table[A108086[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 02 2022 *)
PROG
(Magma)
A108086:= func< n, k | (-1)^Floor((n-k+1)/2)*Binomial(n, k) >;
[A108086(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2022
(SageMath)
def A108086(n, k): return (-1)^int((n-k+1)/2)*binomial(n, k)
flatten([[A108086(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Gerald McGarvey, Jun 05 2005
STATUS
approved