Special case of Theorem 33 of Krathenthaler's Complement Notation: R(a,k) denotes the rising factorial a(a+1)...(a+k-1) Theorem: Let A(m) be the (m+1) by (m+1) matrix binomial(4 + 2 i + 2 j, 2 + i + j) whose (i,j) entry is a(i,j):=, ---------------------------------- i + j + 3 (0<=i,j<=m) . Let b(m):= 2 R(1, m) R(3, m) -----------------, . 2 R(2, m) Then [det A(m)]/[det A(m-1)] = b(m) and hence det [A(m)]=b(1)b(2)...b(m) . Rigorous Proof: Let c(m,n) be n (1/4) R(-m, n) R(m + 3, n) R(1, m) R(5/2, m) ---------------------------------------------, . m R(1, n) R(5/2, n) (1/4) R(-m, m) R(m + 3, m) Note that c(m,m)=1, c(m,n)=0, for n>m . I claim that (1) Sum(c(k,n)*a(m,n),n=0..m)=0 for 0<=k=0 . These two statements imply the theorem since (1) is the defining property for the cofactors of the m-th row up to normalization, and with the normalization such that c(m,m)=1, which is tantamount to dividing the cofactors of the m-th row by the cofact\ or of the (m,m) entry, which is det(A(m-1)), the left of (2) is det[A(m)]/det[A(m-1)]. According to EKHAD, the left side of (1) is annihilated by the operator -2 (2 m + 5) (m + 1) + (m + k + 4) (m - k + 1) M the certificate being, 2 (2 n + 3) n, . This implies that indeed the sum is zero for k