Genocchi number: Difference between revisions
From HandWiki
(over-write) |
(No difference)
|
Latest revision as of 18:51, 6 February 2024
Short description: Mathematical sequence of integers
In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
- [math]\displaystyle{ \frac{-2t}{1+e^{-t}}=\sum_{n=0}^\infty G_n\frac{t^n}{n!} }[/math]
The first few Genocchi numbers are 0, −1, −1, 0, 1, 0, −3, 0, 17 (sequence A226158 in the OEIS), see OEIS: A001469.
Properties
- The generating function definition of the Genocchi numbers implies that they are rational numbers. In fact, G2n+1 = 0 for n ≥ 1 and (−1)nG2n is an odd positive integer.
- Genocchi numbers Gn are related to Bernoulli numbers Bn by the formula
- [math]\displaystyle{ G_{n}=2 \,(1-2^n) \,B_n. }[/math]
Combinatorial interpretations
The exponential generating function for the signed even Genocchi numbers (−1)nG2n is
- [math]\displaystyle{ t\tan \left(\frac{t}{2} \right)=\sum_{n\geq 1} (-1)^n G_{2n}\frac{t^{2n}}{(2n)!} }[/math]
They enumerate the following objects:
- Permutations in S2n−1 with descents after the even numbers and ascents after the odd numbers.
- Permutations π in S2n−2 with 1 ≤ π(2i−1) ≤ 2n−2i and 2n−2i ≤ π(2i) ≤ 2n−2.
- Pairs (a1,...,an−1) and (b1,...,bn−1) such that ai and bi are between 1 and i and every k between 1 and n−1 occurs at least once among the ai's and bi's.
- Reverse alternating permutations a1 < a2 > a3 < a4 >...>a2n−1 of [2n−1] whose inversion table has only even entries.
See also
References
- Weisstein, Eric W.. "Genocchi Number". http://mathworld.wolfram.com/GenocchiNumber.html.
- Richard P. Stanley (1999). Enumerative Combinatorics, Volume 2, Exercise 5.8. Cambridge University Press . ISBN:0-521-56069-1
- Gérard Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Seminaire de Théorie des Nombres de Bordeaux, Volume 11 (1981-1982)
- Serkan Araci, Mehmet Acikgoz, Erdoğan Şen, Some New Identities of Genocchi Numbers and Polynomials
![]() | Original source: https://en.wikipedia.org/wiki/Genocchi number.
Read more |