In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation
2
t
e
t
+
1
=
∑
n
=
1
∞
G
n
t
n
n
!
The first few Genocchi numbers are 1, −1, 0, 1, 0, −3, 0, 17 (sequence A036968 in the OEIS), see A001469.
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
G
n
=
2
(
1
−
2
n
)
B
n
.
There are two cases for
G
n
.
1.
B
1
=
−
1
/
2
from
A027641 /
A027642
G
n
1
= 1, -1, 0, 1, 0, -3 =
A036968, see
A224783
2.
B
1
=
1
/
2
from
A164555 /
A027642
G
n
2
= -1, -1, 0, 1, 0, -3 =
A226158(n+1). Generating function:
−
2
1
+
e
−
t
.
A226158 is an autosequence (a sequence whose inverse binomial transform is the signed sequence) of the first kind (its main diagonal is 0's = A000004). An autosequence of the second kind has its main diagonal equal to the first upper diagonal multiplied by 2. Example: A164555 / A027642.
− A226158 is included in the family:
The rows are respectively A198631(n) / A006519(n+1), − A226158, and A243868.
A row is 0 followed by n (positive) multiplied by the preceding row. The sequences are alternatively of the second and the first kind.
It has been proved that −3 and 17 are the only prime Genocchi numbers.
The exponential generating function for the signed even Genocchi numbers (−1)nG2n is
t
tan
(
t
2
)
=
∑
n
≥
1
(
−
1
)
n
G
2
n
t
2
n
(
2
n
)
!
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.
