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Recursive Formula Involving a Summation

Posted 9 years ago

Hello I am trying to figure out how to correctly input this formula to find any number of coefficients I would like but somewhere along the way I seem to run into trouble. Could anyone tell me where I am going wrong? I thought using the RSolve function would easily solve this for me but all it gives me back is what I put in.

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This is the format I am using, but it won't give me anything meaningful output:

RSolve[{a[n] == (n!/(\[Lambda] - \[Lambda]^n))*
    Sum[(a[n - k]*\[Lambda]^(n - k)*(-1)^k)/(k!*(n - 2 k)!), {k, 1, 
      Floor[n/2]}], a[n] == 1}, a[n], n]

Note that a[1]=1.

POSTED BY: Brian Razin

Asking for a closed form solution may be too much. You can calculate values:

a[1] = 1;
a[n_] := a[n] = (n!/(\[Lambda] - \[Lambda]^n))*
    Sum[(a[n - k]*\[Lambda]^(n - k)*(-1)^k)/(k!*(n - 2 k)!), {k, 1, 
      Floor[n/2]}];
TableForm@Table[Simplify@{n, a[n]}, {n, 1, 10}]
POSTED BY: Gianluca Gorni
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