A generalization of Dixon's identity. P. Bala, July 18 2016. Let D(n) = Sum_{k = 0..n} (-1)^k*C(n,k)^3. Clearly, by symmetry of the binomial coefficients we have D(2*n + 1) = 0. Dixon's identity is the result D(2*n) = (-1)^n*(3*n)!/n!^3. We offer a little generalization. Proposition. Let r be a nonnegative integer and let D_r(n) = Sum_{k = r..n} (-1)^k*C(k,r)^3*C(n,k)^3 (when r > n the empty sum is taken to have the value zero) so that D_0(n) = D(n). Then D_r(n) = (-1)^r*C(n,r)^3*D(n - r). (1) Proof. One of the proofs of Dixon's identity (see [1], for example) begins by showing that D(n) is equal to the constant term in the expansion of (1- x/y)^n*(1 - y/z)^n*(1 - z/x)^n as follows. From the binomial expansion (1 - x)^n = Sum_{i = 0..n} (-1)^i*C(n,i)*x^i (2) we find (1- x/y)^n*(1 - y/z)^n*(1 - z/x)^n = Sum_{0 <= i,j,k <= n} (-1)^(i+j+k)*C(n,i)*C(n,j)*C(n,k)*x^(i-k)*y^(j-i)*z^(k-j). The constant term in the sum is the coefficient of x^0y^0z^0 and occurs when i = j = k and thus has the value Sum_{i = 0..n} (-1)^i*C(n,i)^3 = D(n). Let now 0 <= r <= n be an integer. Differentiate (2) r times and divide the result by r! to find (-1)^r*C(n,r)*(1 – x)^(n-r) = Sum_{i = r..n} (-1)^i*C(i,r)*C(n,i)*x^(i-r). It follows that (-1)^r*C(n,r)^3*(1 – x/y)^(n-r)*(1 - y/z)^(n-r)*(1 – z/x)^(n-r) = Sum_{r <= i,j,k <= n} (-1)^(i+j+k)*C(i,r)*C(j,r)*C(k,r)*C(n,i)*C(n,j)*C(n,k)*x^(i-k)*y^(j-i)*z^(k-j). (3) We equate the constant term on both sides of (3). By our above result, the constant term on the left-hand side of (3) equals (-1)^r*C(n,r)^3*D(n - r): on the right-hand side the constant term occurs when i = j = k and has the value Sum_{i = r..n} (-1)^i*C(i,r)^3*C(n,i)^3. We conclude that D_r(n) = Sum_{i = r..n} (-1)^i*C(i,r)^3*C(n,i)^3 = (-1)^r*C(n,r)^3*D(n - r). REFERENCES [1] James Ward, 100 years of Dixon's Identity, Irish Mathematical Society Bulletin (27): 46–54, 1991. available online at http://www.maths.tcd.ie/pub/ims/bull27/bull27_46-54.pdf ------------------------------------------------------------------------------------------------------------------------------