Genocchi number

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In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation


\frac{2t}{e^t+1}=\sum_{n=1}^{\infty} G_n\frac{t^n}{n!}.

The first few Genocchi numbers are 1, −1, 0, 1, 0, −3, 0, 17 (sequence A036968 in OEIS), see OEISA001469.

Properties[edit]


 G_{n}=2 \,(1-2^n) \,B_n.

There are two cases for G_n.

1. B_1 = -1/2     from OEISA027641 / OEISA027642
G_{n_{1}} = 0, 1, -1, 0, 1, 0, -3 = 0 followed by OEISA036968, see OEISA224783
2. B_1 = 1/2     from OEISA164555 / OEISA027642
G_{n_{2}} = 0, -1, -1, 0, 1, 0, -3 = OEISA226158. Generating function: \frac{-2t}{1-e^{-t}} .

OEISA226158 is an autosequence (a sequence whose inverse binomial transform is a signed sequence) of the first kind (its main diagonal is 0's = OEISA000004). An autosequence of the second kind has its main diagonal equal to the first upper diagonal multiplied by 2. Example: OEISA164555 / OEISA027642.

OEISA226158 is included in the family:

... ... 1 1/2 0 -1/4 0 1/2 0 -17/8 0 31/2
... 0 1 1 0 -1 0 3 0 -17 0 155
0 0 2 3 0 -5 0 21 0 -153 0 1705

The rows are respectively OEISA198631(n) / OEISA006519(n+1), −OEISA226158, and OEISA243868.

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.

Combinatorial interpretations[edit]

The exponential generating function for the signed even Genocchi numbers (−1)nG2n is


 t\tan(\frac{t}{2})=\sum_{n\geq 1} (-1)^n G_{2n}\frac{t^{2n}}{(2n)!}

They enumerate the following objects:

  • 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.

See also[edit]

References[edit]

BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY