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Prove, for any integer n:
[tex]\sum_{0\,\leq\,m\,< n/2}\,(-1)^m(n - 2m)^n\,^nC_m\ =\ 2^{n-1}\,n![/tex]
for example, 7^{7} - 5^{7}7 + 3^{7}21 - 1^{7}35
= 823543 - 546875 + 45927 - 35 = 304560 = 64 times 5040
[tex]\sum_{0\,\leq\,m\,< n/2}\,(-1)^m(n - 2m)^n\,^nC_m\ =\ 2^{n-1}\,n![/tex]
for example, 7^{7} - 5^{7}7 + 3^{7}21 - 1^{7}35
= 823543 - 546875 + 45927 - 35 = 304560 = 64 times 5040