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ISI B.Stat Entrance Exam
2006 ISI B.Stat Entrance Exam
10
10
Part of
2006 ISI B.Stat Entrance Exam
Problems
(1)
f(0)=1,f(1)=0, f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)
Source: ISI(BS) 2006 #10
6/8/2012
Consider a function
f
f
f
on nonnegative integers such that
f
(
0
)
=
1
,
f
(
1
)
=
0
f(0)=1, f(1)=0
f
(
0
)
=
1
,
f
(
1
)
=
0
and
f
(
n
)
+
f
(
n
−
1
)
=
n
f
(
n
−
1
)
+
(
n
−
1
)
f
(
n
−
2
)
f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)
f
(
n
)
+
f
(
n
−
1
)
=
n
f
(
n
−
1
)
+
(
n
−
1
)
f
(
n
−
2
)
for
n
≥
2
n \ge 2
n
≥
2
. Show that
f
(
n
)
n
!
=
∑
k
=
0
n
(
−
1
)
k
k
!
\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}
n
!
f
(
n
)
=
k
=
0
∑
n
k
!
(
−
1
)
k
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