%I #18 Sep 17 2024 20:40:50

%S 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,

%T 0,-5,0,1,0,0,0,0,6,0,-6,0,1,0,0,0,-1,0,10,0,-7,0,1,0,0,0,0,-4,0,15,0,

%U -8,0,1,0,0,0,0,0,-10,0,21,0,-9,0,1,0,0,0,0,1,0,-20,0,28,0,-10,0,1

%N Riordan array (1, x*(1-x^2)).

%C T(2n,n) is a signed aerated version of C(2n,n).

%C Inverse is A124305.

%H G. C. Greubel, <a href="/A124304/b124304.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

%F T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.

%F T(n, k) = C(k,(n-k)/2)*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.

%F Sum_{k=0..n} T(n, k) = A050935(n+2).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).

%F T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).

%F From _G. C. Greubel_, Aug 18 2023: (Start)

%F T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).

%F T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).

%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).

%F G.f.: 1/(1 - x*y*(1-x^2)). (End)

%e Triangle begins

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 0, -1, 0, 1;

%e 0, 0, -2, 0, 1;

%e 0, 0, 0, -3, 0, 1;

%e 0, 0, 1, 0, -4, 0, 1;

%e 0, 0, 0, 3, 0, -5, 0, 1;

%e 0, 0, 0, 0, 6, 0, -6, 0, 1;

%t A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;

%t Table[A124304[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 18 2023 *)

%o (Magma)

%o A124304:= func< n,k | (&+[(-1)^j*Binomial(k,k-j)*Binomial(k,n-k-j) : j in [0..n]]) >;

%o [A124304(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Aug 18 2023

%o (SageMath)

%o def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2

%o flatten([[A124304(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Aug 18 2023

%Y Cf. A014021 (diagonal sums), A050935 (row sums), A124305 (inverse).

%Y Cf. A002054, A079977, A126869, A138364, A176971.

%K easy,sign,tabl

%O 0,13

%A _Paul Barry_, Oct 25 2006