Pb 1. Nombres de Catalan.

a b a < b
(O, (−→ i , −→ j)) x
y
M x(M) = y(M)
M y(M)≤x(M)
M y(M)< x(M)
(M0, M1,· · · Mp)∀k∈J0, p −1K,−−−−−−→ MkMk+1 ∈ {−→ i , −→ j}
Mkp
M0Mp
Pa,b (a, a)
(b, b)
Ca,b Pa,b
C0
a,b Pa,b
Γ=(M0, M1,· · · Mp)∈ Ca,b m(Γ) x(Mk)>0Mk
n∈N∗cnC0,n c0= 1
Γ∈ C0,5
m(Γ)
n∈N∗
P0,n
P0,n
c1c2
Ca,b ckk
m∈N∗C0
0,m ck
k
∀n∈N∗, cn=
n
X
m=1
cm−1cn−m
C0,5
∀n∈N, cn+1 =
n
X
k=0
ckcn−k
cn
cn=2n
n
n+ 1
n
an=2n
n
n+ 1, Sn=
n
X
k=0
akan−k, Tn=
n
X
k=0
kakan−k
a0= 1
2Tn=nSnTn+1 +Sn+1 =n+3
2Sn+1
(k+ 2)ak+1 = 2(2k+ 1)akTn+1 +Sn+1 =an+1 + 4Tn+ 2Sn
Sn=an+1 Sn+1 =an+2

x(bnxc)n∈N∗
V(x)
M(x)
V(x) = {bnxc, n ∈N∗}, M(x) = {nx, n ∈N∗}
s r
1
s+1 r= 1
V(s)V(r)N∗
j k m j =bkrc=bmsc
j < k +m < j + 1 V(s)∩V(r) = ∅
j∈N∗j /∈V(r)k∈N
kr < j j + 1 ≤(k+ 1)r
j k m
kr < j j + 1 ≤(k+ 1)r ms < j j + 1 ≤(m+ 1)s
k+m < j < k +m+ 1
M(1
r)M(1 s)
j∈N∗
x∈M(1
r)x≤j r,x∈M(1 s)x≤j r
Q
]x∈M(1
r)∪M(1 s)x≤j r=bjsc
M(1
r)∪M(1 s)
(an)n∈N∗
(xn)n∈N∗(yn)n∈N∗
∀n∈N∗,
xn= max nak
k, k ∈J1, nKo
yn= min ak+ 1
k, k ∈J1, nK
(xn)n∈N∗(yn)n∈N∗
(an)n∈N∗
α > 0an=bnαcn∈N∗
xn< ynn∈N∗
xn< ynn∈N∗
(xn)n∈N∗(yn)n∈N∗
α
α an=bnαcn∈N∗
ak= 2k−1k∈N∗
Πp∆
P(Π) Π
∆
a b Π
D(a, b)
a1a2a3a4

4
6
Π
δ δ0O
f:(δ→δ0
a7→ D(O, a)∩δ0
d
O nOO
Π\ {O}O
nO
O
n
n+ 1
n2+n+ 1