bit of editing wanted : are these equivalent?
benoit
abcloitre at wanadoo.fr
Sat Dec 6 22:56:52 CET 2003
> I hit on it using something different (or does it take a trained eye
> to see the equivalence with A002425 ?)
>
> Numerator[ CoefficientList[Series[1/(1 + Cosh[Sqrt[x]]), {x, 0, 16}],
> x]*Range[0, 16]! ]
>
> 1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581,
> 4722116521,..
>
> or, equivalently,Numerator[ CoefficientList[Series[1/(1 + Cosh[x]),
> {x, 0, 16}], x]*Range[0, 16]! ]
>
> 1, 0, -1, 0, 1, 0, -17, 0, 31, 0, -691, 0, 5461, 0, -929569, 0, 3202291
>
> where the denominators are
> http://www.research.att.com/projects/OEIS?Anum=A006519
> (Highest power of 2 dividing n.)
>
> (have I stumbled upon an EGF for A002425?)
>
>
Yes that's an interesting equivalence. Let (A(n)) having g.f.
x/(1+cosh(x)) and (B(n)) having g.f. 2/(1+exp(x))
then A(n)+n*B(n)=0
check :
? A(n)=polcoeff(x/(1+cosh(x)),n)
? B(n)=polcoeff(2/(1+exp(x)),n)
? for(n=1,100,print1(C(n)+n*E(n),","))
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.........
Benoit
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