Concrete mathematics : a foundation for computer science

6.2 EULERIAN NUMBERS 255
Eulerian numbers are useful primarily because they provide an unusual
connection between ordinary powers and consecutive binomial coefficients:
xn = F(L)(“L”>, integern>O.
(This is “Worpitzky’s identity” [308].) For example, we have
x2 -
x3 =
(1)+(T))
(;)+qy)+(y’),
(;)+ll(x;')+11(Xfi2)+(X;3),
and so on. It’s easy to prove (6.37) by induction (exercise 14).
Incidentally, (6.37) gives us yet another way to obtain the sum of the
first n squares: We have k2 = ($(i) + (f) (“i’) = (i) + (ki’), hence
12+22+...+n2 = ((;)+(;)+-.+(;))+((;)+(;)+.-+(";'))
= ("p) + ("f2) = ;(n+l)n((n-l)+(n+2)).
The Eulerian recurrence (6.35) is a bit more complicated than the Stirling
recurrences (6.3) and (6.8), so we don’t expect the numbers (L) to satisfy as
many simple identities. Still, there are a few:
(t) = g (n:‘)(m+l -k)“(-llk;
-!{Z} = G(E)(n*m)’
(;) = $ {;}(“,“)(-l)nPk-mk!
(6.38)
(6.39)
(6.40)
If we multiply (6.39) by znPm and sum on m, we get x,, { t}m! zn-“’ =
tk (c) (z + 1) k. Replacing z by z - 1 and equating coefficients of zk gives
(6.40). Thus the last two of these identities are essentially equivalent. The
first identity, (6.38), gives us special values when m is small:
(i) = 1; (I) = 2n-n-l; (1) = 3”-(n+l)Z”+(n:‘) .