Presumably this is a well-known identity. Does it have a name?pic.twitter.com/tsMBGLZGAk
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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?
Did I really get that wrong? Sorry. OK, so you proceed inductively: first, write x^(n-1) as a sum of falling factorials (with Stirling number coefficients), and then get x^n from that sum by multiplying each of its constituent summands by the appropriate [(x - n) + n] factor.
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