Genocchi Number

Author

Eric W. Weisstein
August 30, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/GenocchiNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GenocchiNumber.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

GenocchiG[n_]:=2(1-(2^n))BernoulliB[n]

Enumeration

GenocchiG/@Range[20]

Primes

2Position[GenocchiG/@Range[2,100,2],_?PrimeQ]//Flatten//Timing

Select[Table[2(1-(2^(2n)))BernoulliB[2n],{n,1000}],PrimeQ]//Timing

Select[Table[2(1-(2^(2n)))BernoulliB[2n],{n,2000}],PrimeQ]//Timing

Position[Table[2(1-(2^(2n)))BernoulliB[2n],{n,2000,4000}],_?PrimeQ]//Flatten//Timing

Position[Table[2(1-(2^(2n)))BernoulliB[2n],{n,4000,5000}],_?PrimeQ]//Flatten//Timing

GenocchiG[8]

Series

Series[((2t)/((E^t)+1)),{t,0,10}]

c=CoefficientList[Normal[%],t]

c Range[0,10]!

V6

<<DiscreteMath`RSolve`

SeriesTerm[((2t)/((E^t)+1)),{t,0,n}]

FullSimplify[n! %]

Table[n EulerE[-1+n,0],{n,10}]

V7

SeriesCoefficient[((2t)/((E^t)+1)),{t,0,n}]//InputForm

FullSimplify[((EulerE[-1+n,0])/((-1+n)!))]

FullSimplify[2(1-(2^n))BernoulliB[n]-((EulerE[-1+n,0])/(Gamma[n])),Mod[n,2]==1]

FunctionExpand[((EulerE[-1+n,0])/(Gamma[n])),Mod[n,2]==1]

Table[-2 (-1+(2^n)) BernoulliB[n]-((EulerE[-1+n,0])/(Gamma[n])),{n,1,20,2}]

GeneratingFunction[n EulerE[-1+n,0],n,x]//InputForm