Difference between revisions of "Genocchi numbers"

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(Properties)
(Properties)
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=Properties=
 
=Properties=
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<strong>Proposition:</strong> The following values hold for the Genocchi numbers:
 
$$G_1=1, G_3=G_5=G_7=G_9=G_11=\ldots=0.$$
 
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<strong>Proof:</strong> █
 
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<strong>Proposition:</strong> The following formula holds:
 
<strong>Proposition:</strong> The following formula holds:

Revision as of 16:40, 30 April 2015

The Genocchi numbers $G_n$ are given by the generating function $$\dfrac{2t}{e^t+1} = \displaystyle\sum_{k=0}^{\infty} G_n \dfrac{t^n}{n!}.$$

Properties

Proposition: The following formula holds: $$G_{2n}=2(1-2^{2n})B_{2n}= 2nE_{2n-1}(0),$$ where $G_{2n}$ denotes Genocchi numbers, $B_{2n}$ denotes Bernoulli numbers, and $E_{2n-1}$ denotes an Euler polynomial.

Proof: