Recurrence Table for a specific Recurrence

Question

I wanted to know how can one find an arbitrary number of terms of the non-linear recurrence (using software, like maple)such as $$a_{2n-1}=-\sum_{j=1}^{n}{n+j-1\choose 2j-2}a_{n+j-2}, \ a_{0}=-1$$ $$a_{2n}=-\sum_{j=1}^{n}{n+j\choose 2j-1}a_{n+j-1}.$$ I have tried something like the following code into maple, yet it doesn't yield anything useful,

a := n -> rsolve({a(0) = -1, a(2*n) = -sum(binomial(n + j, 2*j - 1)*a(n + j - 1), j = 1 .. n), a(2*n - 1) = -sum(binomial(n + j - 1, 2*j - 2)*a(n + j - 2), j = 1 .. n)}, a(n), makeproc);.

Edit:I know that there is the Mathematica recurrence table, but don't know how to get it working for this particular example. I have also tried the Rsolve as shown above, it works for some examples like the second-order linear recurrences but not for this one.

The link for recurrence table:-https://reference.wolfram.com/language/ref/RecurrenceTable.html.

Answer 1

You could make different definitions of a[n] for n even and n odd. At the same time it is beneficial to use memorization to prevent the calculation of the same a[n] multiple times. E.g:

Clear["Global`*"]
a[0] = 1;
a[n_?OddQ] := 
  a[n] = (m = (n + 1)/2; -Sum[
      Binomial[m + j - 1, 2 j - 2] a[m + j - 2], {j, m}]);

a[n_?EvenQ] := 
  a[n] = (m = 
     n/2; -Sum[Binomial[m + j, 2 j - 1] a[m + j - 1], {j, m}]);


Table[a[i], {i, 0, 10}]

(* {1, -1, 2, -5, 14, -42, 132, -429, 1430, -4862, 16796} *)