OFFSET
0,13
COMMENTS
T(2n,n) is a signed aerated version of C(2n,n).
Inverse is A124305.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
FORMULA
T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.
T(n, k) = C(k,(n-k)/2)*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A050935(n+2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).
T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).
From G. C. Greubel, Aug 18 2023: (Start)
T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).
T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).
G.f.: 1/(1 - x*y*(1-x^2)). (End)
EXAMPLE
Triangle begins
1;
0, 1;
0, 0, 1;
0, -1, 0, 1;
0, 0, -2, 0, 1;
0, 0, 0, -3, 0, 1;
0, 0, 1, 0, -4, 0, 1;
0, 0, 0, 3, 0, -5, 0, 1;
0, 0, 0, 0, 6, 0, -6, 0, 1;
MATHEMATICA
A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;
Table[A124304[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)
PROG
(Magma)
A124304:= func< n, k | (&+[(-1)^j*Binomial(k, k-j)*Binomial(k, n-k-j) : j in [0..n]]) >;
[A124304(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023
(SageMath)
def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2
flatten([[A124304(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Oct 25 2006
STATUS
approved