# [seqfan] Re: G.f. for trees with degree at most 3

Georgi Guninski guninski at guninski.com
Mon May 24 13:45:13 CEST 2010

```On Mon, May 24, 2010 at 09:38:27AM +0800, Douglas McNeil wrote:
>
> Seems to work for me:
>
> sage: gf = ((1-x)*(2-x-x**2) - (2-x+x**2)*(1-2*x-x**2)**(1/2)) / (3*x**3)
> sage: c = taylor(gf, x, 0, 12).coefficients()
> sage: c
> [[1, 0], [1, 1], [3/2, 2], [8/3, 3], [5, 4], [39/4, 5], [469/24, 6],
> [40, 7], [333/4, 8], [1405/8, 9], [5995/16, 10], [807, 11], [42055/24,
> 12]]
> sage: sq = [a*factorial(b) for a,b in cc]
> sage: sq
> [1, 1, 3, 16, 120, 1170, 14070, 201600, 3356640, 63730800, 1359666000,
> 32212857600, 839350512000]
>
> Think you've changed a sign in "2-2+x**2".
>
>

this seems to hold up to 3000 terms:

a[n+1]=(  -a[n-1]*a[n]+   -  ( -3*a[n]**2 + (2/3)*a[n-2]*a[n]*n+
(-4/3)*a[n-1]*a[n]*n+ (-4/3)*a[n]**2*n+ (-1/3)*a[n-2]*a[n]*n**2+
(-2/3)*a[n-1]*a[n]*n**2)) / (a[n-1]+ (-1/3)*a[n] -2*a[n-2]*n+
2*a[n-1]*n+a[n-2]*n**2)

for a[n] == A003692

```

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