Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
PeterBurbery`Combinatorics`
SecantNumber  |
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Details and Options
Examples
(1)
Basic Examples
(1)
Make a table of secant numbers:
In[1]:=
 | SecantNumber |
Out[1]=
{1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525,69348874393137901,15514534163557086905,4087072509293123892361,1252259641403629865468285,441543893249023104553682821,177519391579539289436664789665,80723299235887898062168247453281,41222060339517702122347079671259045,23489580527043108252017828576198947741,14851150718114980017877156781405826684425,10364622733519612119397957304745185976310201}
Here are some different ways of computing the sequence of secant numbers.
In[2]:=
Take[Range[0,32]!*CoefficientList[Series[Sec[x],{x,0,32}],x],{1,32,2}](*RobertG.Wilsonv,Apr232006*)
Out[2]=
{1,1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525,69348874393137901,15514534163557086905,4087072509293123892361,1252259641403629865468285,441543893249023104553682821}
In[3]:=
Table[Abs[EulerE[2n]],{n,0,10}](*RayChandler,Mar202007*)
Out[3]=
{1,1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525}
In[4]:=
a[n_]:=If[n<0,0,With[{m=2n},m!SeriesCoefficient[Sec[x],{x,0,m}]]];(*MichaelSomos,Nov222013*)
In[5]:=
Table[a[n],{n,1,10,1}]
Out[5]=
{1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525}
In[6]:=
a[n_]:=If[n<0,0,With[{m=2n+1},m!SeriesCoefficient[InverseGudermannian[x],{x,0,m}]]];(*MichaelSomos,Nov222013*)
In[7]:=
Table[a[n],{n,1,10,1}]
Out[7]=
{1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525}
In[8]:=
a[n_]:=Sum[Sum[Binomial[k,m](-1)^(n+k)/(2^(m-1))Sum[Binomial[m,j]*(2j-m)^(2n),{j,0,m/2}](-1)^(k-m),{m,0,k}],{k,1,2n}];Table[a[n],{n,0,16}](*Jean-FrançoisAlcover,Jun262019,afterVladimirKruchinin*)
Out[8]=
{0,1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525,69348874393137901,15514534163557086905,4087072509293123892361,1252259641403629865468285,441543893249023104553682821,177519391579539289436664789665}
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""