Английская Википедия:Genocchi number
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Шаблон:Short description In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
- <math>
\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 Шаблон:OEIS, see Шаблон:OEIS2C.
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>
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>
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.
Primes
The only known prime numbers which occur in the Genocchi sequence are 17, at n = 8, and -3, at n = 6 (depending on how primes are defined). It has been proven that no other primes occur in the sequence
See also
References
- Шаблон:MathWorld
- Richard P. Stanley (1999). Enumerative Combinatorics, Volume 2, Exercise 5.8. Cambridge University Press. Шаблон:ISBN
- 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