BINOMNI KOEFICIJENTI
BINOMNI KOEFICIJENTI
BINOMNI KOEFICIJENTI
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<strong>BINOMNI</strong> TEOREM<br />
Teorem 9 (Vandermonedeova konvolucija)<br />
(<br />
(i) n<br />
)( m<br />
) (<br />
0 r + n m<br />
(<br />
1)(<br />
r−1)<br />
+<br />
n<br />
)( m<br />
(<br />
2 r−2)<br />
+ . . . +<br />
n m<br />
∑<br />
r)(<br />
0)<br />
=<br />
r<br />
( n m<br />
) (<br />
k=0 k)(<br />
r−k = m+n<br />
)<br />
( r<br />
(ii) n<br />
)( m<br />
) (<br />
0 0 + n<br />
)( m<br />
) (<br />
1 1 + n<br />
)( m<br />
) (<br />
2 2 + . . . + n<br />
)( m<br />
) ∑<br />
n n = n<br />
( n<br />
)( m<br />
(<br />
k=0 k k)<br />
=<br />
m+n<br />
)<br />
n<br />
Specijalno za m = n<br />
( ) 2 ( )<br />
n∑ n 2n<br />
= .<br />
k n<br />
k=0<br />
Dokaz: algebarski<br />
Dva polinoma se množe na slijedeći način:<br />
(a 0 + a 1x + a 2x 2 + . . . + a nx n )(b 0 + b 1x + b 2x 2 + . . . + b nx n ) = a 0b 0 + (a 1b 0 + a 0b 1)x<br />
+(a 0b 2 + a 1b 1 + a 2b 0)x 2 + (a 0b 3 + a 1b 2 + a 2b 1 + a 3b 0)x 3 + . . . + a nb mx n+m<br />
n+m<br />
∑<br />
= (a 0b r + a 1b r−1 + a 2b r−2 + . . . + a rb 0)x r<br />
r=0<br />
() 21. studenog 2011. 19 / 27