Presumably this is a well-known identity. Does it have a name?pic.twitter.com/tsMBGLZGAk
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For of course each S(n, k) is a sum of all the k-fold products of the numbers from 0 to n -- that is, S(n, k) is the kth elementary symmetric polynomial in n+1 variables, evaluated at 0, 1, ... , n. And of course those terms whose factors include 0 contribute nothing to the sum.
So the sum that appears on the left-hand side of your identity is indeed S(n, k). Which means that the only thing left is to show how the S(n, k) relate to the Stirling numbers. And it's clear from their definition that they relate the falling-factorial basis to the power basis.
Thank you! You were not obscure at all: I was just missing the obvious, as I suspected.
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