info/mathtrip/src/ids3.tex


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%This command provides the text of the identities displayed at the
%top of page 3
%
%The command has one parameter:
%        1) The width of the text
\newcommand\TTwoIdentities[1]{%
   \parbox[t]{#1}{%
      \TTwoFormulaeFontSize
      \begin{DisplayFormulae}{38}{\SpaceBeforeFormula}{\TTwoInterlineFormulae}{\BigChar}{\StyleBold}
         \Fm{\Cycle{n+1}{m+1} = \sum_k \Cycle{n}{k} \binom{k}{m} = \sum_{k=0}^n \Cycle{k}{m}
             n^{\underline{n-k}} = n! \sum_{k=0}^n \frac{1}{k!} \Cycle{k}{m}}
         \Fm{\Cycle{x}{x-n} = \sum_{k=0}^n \Euls{n}{k} \binom{x+k}{2n}}
         \Fm{\SousEnsemble{n}{m} = \sum_k \binom{n}{k} \SousEnsemble{k+1}{m+1}(-1)^{n-k}}
         \Fm{\Cycle{n}{m} = \sum_k \Cycle{n+1}{k+1} \binom{k}{m}(-1)^{m-k}}
         \Fm{\SousEnsemble{m+n+1}{m} = \sum_{k=0}^m k  \SousEnsemble{n+k}{k}}
         \Fm{\Cycle{m+n+1}{m} = \sum_{k=0}^m k  (n+k)\Cycle{n+k}{k}}
         \Fm{\binom{n}{m} = \sum_k \SousEnsemble{n+1}{k+1} \Cycle{k}{m} (-1)^{m-k}}
         \Fm{(n-m)!\binom{n}{m} = \sum_k \Cycle{n+1}{k+1} \SousEnsemble{k}{m} (-1)^{m-k}\MathRemark{\text{for }n \geq m}}
         \Fm{\SousEnsemble{n}{n-m} = \sum_k \binom{m-n}{m+k} \binom{m+n}{n+k}\Cycle{m+k}{k}}
         \Fm{\Cycle{n}{n-m} = \sum_k \binom{m-n}{m+k} \binom{m+n}{n+k}\SousEnsemble{m+k}{k}}
         \Fm{\SousEnsemble{n}{\ell+m}\binom{\ell+m}{\ell} = \sum_k \SousEnsemble{k}{\ell} \SousEnsemble{n-k}{m} \binom{n}{k}}
         \Fm{\Cycle{n}{\ell+m}\binom{\ell+m}{\ell} = \sum_k \Cycle{k}{\ell} \Cycle{n-k}{m} \binom{n}{k}}
      \end{DisplayFormulae}
   }%
}