Moments of Catalan Triangle Numbers
In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, ge...
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Capítulo de libro |
| Lenguaje: | inglés |
| Publicado: |
IntechOpen
2020
|
| Materias: | |
| Acceso en línea: | https://www.intechopen.com/chapters/71899 |
| Etiquetas: |
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| Sumario: | In this chapter, we consider the Catalan numbers,
C
n
=
1
n
+
1
2
n
n
, and two of their generalizations, Catalan triangle numbers,
B
n
,
k
and
A
n
,
k
, for
n
,
k
∈
N
. They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums:
∑
k
=
1
n
k
m
B
n
,
k
j
,
∑
k
=
1
n
+
1
2
k
−
1
m
A
n
,
k
j
,
for
j
,
n
∈
N
and
m
∈
N
∪
0
. We present their closed expressions for some values of
m
and
j
. Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section. |
|---|---|
| ISBN: | 978-1-83968-050-2 |
| DOI: | 10.5772/intechopen.92046 |