# Coefficients of Knuth's “Convolution Polynomials”

Donald E. Knuth gives in his paper Convolution Polynomials the code

F[n_, x_] := Sum[f[n, j] x^j, {j, 0, n}]/n!
conv[n_] := LogicalExpand[Series[F[n, x + y], {x, 0, n}, {y, 0, n}]
== Series[Sum[F[k, x] F[n - k, y], {k, 0, n}], {x, 0, n}, {y, 0, n}]]
Solve[Table[conv[n], {n, 0, 4}], [Flatten[Table[f[i, j], {i, 0, 4},{j, 0, 4}]]]]


and then writes: "Mathematica replies that the $F$'s are either identically zero or the coefficients of $F_n(x)$ satisfy ..." How can I get the table following this statement?

• Try fixing the syntax (typo): There a [ in front of Flatten and an extra ] at the end that should be removed. – Michael E2 Jan 6 '17 at 11:33

As Michael notes, deleting the stray brackets will make it work. A better method, however, would be to use SolveAlways[]:
F[n_, x_] := Sum[f[n, j]x^j, {j, 0, n}]/n!