Presumably this is a well-known identity. Does it have a name?pic.twitter.com/tsMBGLZGAk
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My proof is neither of these, and I'm delighted to learn of your proofs, which are probably more satisfactory. I simply show that the Stirling numbers (of the right kind) are the coefficients of the "descending factorial" polynomials, by straightforward algebraic-inductive means.
The essence of the argument is almost mindlessly straightforward: x^2 = x(x - 1) + x, x^3 = [x(x - 1)][(x - 2) + 2] + x[(x - 1) + 1], x^4 = [x(x - 1)(x - 2)][(x - 3) + 3] + [x(x - 1)][(x - 2) + 2] + x[(x - 1) + 1], etc., reveals the required role for the Stirling numbers.
I’m afraid the incorrect expression for x^4 is making it hard for me to see the pattern you’re illustrating here. I’m sorry to be so slow. Would you mind giving the expression you intended for x^4?
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