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In [[mathematics]], the '''Genocchi numbers''' G''n'', named after [[Angelo Genocchi]], are a [[sequence (mathematics)|sequence]] of [[integer]]s that satisfy the relation

:<math>
\frac{2t}{e^t+1}=\sum_{n=1}^{\infty} G_n\frac{t^n}{n!}.
</math>

The first few Genocchi numbers are 1, −1, 0, 1, 0, −3, 0, 17 {{OEIS|id=A001469}}.

== Properties ==

* The [[generating function]] definition of the Genocchi numbers implies that they are [[rational number]]s. In fact, G''2n+1'' = 0 for ''n'' ≥ 1 and (−1)''n''G''2n'' is an [[odd number|odd]] positive integer.

* Genocchi numbers ''G''''n'' are related to [[Bernoulli numbers]] ''B''''n'' by the formula

: <math>
G_n=2 \,(1-2^n) \,B_n.
</math>

* It has been proved that −[[3 (number)|3]] and [[17 (number)|17]] are the only [[prime number|prime]] Genocchi numbers.

== Combinatorial interpretations ==

The [[exponential generating function]] for the '''signed even Genocchi numbers''' (−1)''n''G''2n'' is
:<math>
t\tan(\frac{t}{2})=\sum_{n\geq 1} (-1)^n G_{2n}\frac{t^{2n}}{(2n)!}
</math>

They enumerate the following objects:

* [[Permutation]]s in ''S''2''n''−1 with [[Permutation#Ascents.2C_descents_and_runs|descents]] after the even numbers and [[Permutation#Ascents.2C_descents_and_runs|ascents]] after the odd numbers.

* Permutations ''π'' in ''S''2''n''−2 with 1 ≤ ''π''(2''i''−1) ≤ 2''n''−2''i'' and 2''n''−2''i'' ≤ ''π''(2''i'') ≤ 2''n''−2.

* Pairs (''a''1,…,''a''''n''−1) and (''b''1,…,''b''''n''−1) such that ''a''''i'' and ''b''''i'' are between 1 and ''i'' and every ''k'' between 1 and ''n''−1 occurs at least once among the ''a''''i'''s and ''b''''i'''s.

* Reverse [[alternating permutation]]s ''a''1 < ''a''2 > ''a''3 < ''a''4 >…>''a''2''n''−1 of [2''n''−1] whose [[Permutation#Numbering_permutations|inversion table]] has only even entries.

== See also ==

* [[Euler number]]

== References ==

* {{MathWorld|urlname=GenocchiNumber|title=Genocchi Number}}

* [[Richard P. Stanley]] (1999). [http://www-math.mit.edu/~rstan/ec/ ''Enumerative Combinatorics'', Volume 2], Exercise 5.8. [[Cambridge University Press]]. ISBN 0-521-56069-1

* Gérard Viennot, [http://resolver.sub.uni-goettingen.de/purl?PPN320141322_0011/dmdlog41 ''Inteprétations combinatoires des nombres d'Euler et de Genocchi''], Seminaire de Théorie des Nombres de Bordeaux, Volume 11 (1981-1982)

[[Category:Integer sequences]]
[[Category:Factorial and binomial topics]]

Modified Morlet wavelet - Revision history

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