Why is this series given by this Taylor expansion?












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So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:



$$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$



i.e. $N(0) = 1 $,

and $N(1) = 1 + 8 = 9$
etc.




By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:



$$ frac{(1+x)^4}{(1-x)^5}. $$



I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
Does anyone know whether this is a known result? Does that function ring a bell?






sequences-and-series graph-theory taylor-expansion generating-functions







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    $begingroup$


    So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:



    $$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$



    i.e. $N(0) = 1 $,

    and $N(1) = 1 + 8 = 9$
    etc.




    By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:



    $$ frac{(1+x)^4}{(1-x)^5}. $$



    I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
    Does anyone know whether this is a known result? Does that function ring a bell?






    sequences-and-series graph-theory taylor-expansion generating-functions







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:



      $$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$



      i.e. $N(0) = 1 $,

      and $N(1) = 1 + 8 = 9$
      etc.




      By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:



      $$ frac{(1+x)^4}{(1-x)^5}. $$



      I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
      Does anyone know whether this is a known result? Does that function ring a bell?






      sequences-and-series graph-theory taylor-expansion generating-functions







      share|cite|improve this question











      $endgroup$




      So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:



      $$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$



      i.e. $N(0) = 1 $,

      and $N(1) = 1 + 8 = 9$
      etc.




      By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:



      $$ frac{(1+x)^4}{(1-x)^5}. $$



      I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
      Does anyone know whether this is a known result? Does that function ring a bell?






      sequences-and-series graph-theory taylor-expansion generating-functions




      sequences-and-series graph-theory taylor-expansion generating-functions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 2 at 3:57









      Alex Ravsky

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      42.9k32483










      asked Jan 31 at 16:40









      SuperCiociaSuperCiocia

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      295213






















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