[seqfan] Re: Definition of A002190
Alois Heinz
heinz at hs-heilbronn.de
Sun Oct 10 01:03:37 CEST 2010
The offset should be 0 with a(0)=0, a(1)=1, a(2)=1, ...
Maple code:
a:= n-> coeff (series (-ln(BesselJ(0,2*sqrt(x))), x, n+1), x, n)*(n!)^2;
seq (a(n), n=0..30);
gives:
0,1,1,4,33,456,9460,274800,10643745,530052880,32995478376, ...
see also the signed version A101981
Alois
Am 09.10.2010 23:52, schrieb Paul D Hanna:
> Seqfans,
> The definition of A002190 is:
> Sum_{n>=0} a(n)*x^n/n!^2 = -ln(BesselJ(0,2*sqrt(x))).
> [1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, ...]
> (OFFSET 2).
>
> However, by the definition the offset should be 1 and a(1) = 1, since
> -log( Sum_{n>=0} (-x)^n/n!^2 ) = x + x^2/2!^2 + 4*x^3/3!^2 + 33*x^4/4!^2 + 456*x^5/5!^2 + 9460*x^6/6!^2 +...
>
> The question is, should the sequence be changed to include a(1)=1,
> or should the definition be changed to fit the existing sequence, i.e.,
> change the name to:
> Sum_{n>=0} a(n)*x^n/n!^2 = -x-ln(BesselJ(0,2*sqrt(x))).
>
> I would vote for prepending a '1' to the existing sequence and changing OFFSET=1 to fit the definition,
> but I fear changing an old classic sequence without the consensus of SeqFans.
>
> Thanks,
> Paul
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