# [seqfan] G.F. for A245925 Sought

Paul D Hanna pauldhanna at juno.com
Sat Aug 16 18:32:19 CEST 2014

```Seqfans,     Here is an interesting sequence that needs a g.f. with a closed form:

http://oeis.org/A245925
1, -3, 25, -243, 2601, -29403, 344569, -4141875, 50737129, ...

The generating function is given by the binomial series identity:

A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j

= Sum_{n>=0} x^n / (1+x)^(2*n+1) * [ Sum_{k=0..n} C(n,k)^2*(-x)^k ]^2

and the term-by-term formula is:

a(n) = Sum_{k=0..n} Sum_{j=0..2*n-2*k} (-1)^(j+k) * C(2*n-k,j+k)^2 * C(j+k,k)^2.

What is surprising about the terms in A245925 is that they involve perfect squares:

a(2*n) = A245926(n)^2,

a(2*n+1) = (-3)*A245927(n)^2,

where the g.f.s for A245926 and A245927 have a closed form:

A245926: sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) ),

A245927: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ).

Perhaps someone could derive a g.f. with a ~similar closed form for A245925.

Thanks,
Paul
```