R. J. Mathar's conjecture given above is indeed a theorem. Proof: Start from the first formula: a(n) = n!*Sum {k=1..n, (-1)^(n+1)*1/k}. Fold the sum: a(n) = -(-1)^n*H(n)*n! where H(n) is the harmonic number. Substitute the last expression into the recurrence relation, then and use identities (n-1)! = n!/n and H(n-1) = H(n)-1/n as follows -(-1)^(n-2)*(n-1)^2*H(n-2)*(n-2)! - (-1)^(n-1)*(2n-1)*H(n-1)*(n-1)! - (-1)^n*H(n)*n! = -(-1)^n*(n-1)^2*(H(n)-1/n-1/(n-1))*n!/n/(n-1) + (-1)^n*(2n-1)*(H(n)-1/n)*n!/n - (-1)^n*H(n)*n! = -(-1)^n*n!*((n-1)*(H(n)-1/n-1/(n-1))/n - (2n-1)*(H(n)-1/n)/n + H(n)) = -(-1)^n*n!*(H(n) - 1/(n-1) + 1/n^2 - 1/n - H(n)/n + 1/n/(n-1) - 2 H(n) - 1/n^2 + 2/n + H(n)/n + H) = -(-1)^n*n!*(-1/(n-1) + 1/n + 1/n/(n-1)) = -(-1)^n*n!*0 = 0 QED