Mathematical sequence of integers
In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
![{\displaystyle {\frac {2t}{1+e^{t}}}=\sum _{n=0}^{\infty }G_{n}{\frac {t^{n}}{n!}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The first few Genocchi numbers are 0, 1, −1, 0, 1, 0, −3, 0, 17 (sequence A226158 in the OEIS), see OEIS: A001469.
Properties
![{\displaystyle G_{n}=2\,(1-2^{n})\,B_{n}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Combinatorial interpretations
The exponential generating function for the signed even Genocchi numbers (−1)nG2n is
![{\displaystyle t\tan \left({\frac {t}{2}}\right)=\sum _{n\geq 1}(-1)^{n}G_{2n}{\frac {t^{2n}}{(2n)!}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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