EXERCICES D`ALGÈBRE : ÉNONCÉS

∀n∈N\ {0},∀m∈N,06m6nn
m=n n−m
∀n∈N\ {0},n
X
i=0 n
i= 2n.
∀n∈N,∀m∈N,16m6n
n
m=n m−1+n−1 m−1
∀n∈N\ {0}
[n
2]
X
i=0 n
2i=
[n−1
2]
X
i=0 n
2i+ 1 .
l+m
k=
min (l,k)
X
i=max (0,k−m)l
i m k−i.
n(F, G)F G E
F∩G=∅ {F, G}F G E F ∩G=∅
E{1, ..., n} SESn
{1, ..., n} \ E(i, j)Sn
<SE,(i, j)>={SE×<(i, j)> i /∈E j /∈E
SE∪{j}i∈E j /∈E

XSnGX
{1, ..., n} {(i, j)|(i, j)∈X}X
SnGX
Snn−1
n2n
0,1,2...., 2n−2.
2n−2
i<n−1i
2(n−1) −i i < n −1
DnSn
S(X) = ∞
X
n=0
Dn
n!Xn.
Dn=n!−n!
1! +n! 2! +... + (−1)nn! n!.
Dne−1n!
n! = Pn
i=0 Ci
nDn−i.
S(X)eX=1
1−X.
Dn
Sn,k n k
Sn,k
i{1, ..., k}
Sn,k
knn+∞
Bnn
B0= 1 T
T=∞
X
n=0
Bn
n!Xn.

{1, ..., n}k n
Bn=
n
X
k=1
(n−1
k−1)Bn−k.
T T 0=eXT
T(X) = eeX−1.
BnSn,k n
k
G E k
g∈G(g)E g
k=1
|G|X
g∈G|(g)|.
n m 1
nPd|nϕ(d)mn/d
A5
(96)