[seqfan] Re: The Demichel errors, continued

[N. J. A. Sloane]

Olivier's analysis is probably correct, and
Patrick has rounded rational numbers to integers when 
he should not have done.

I propose that the best way to fix the problem
is to throw away the present sequence if
it is wrong (yes, Marc was right, throw away the
bad food), and replace it with two sequences,
one giving the numerators and the other the denominators.

Let me take A013305 as an example.

I'll work with the internal format, for simplicity. Here is what we have now:

%I
%S 1,6,0,0,0,259,19,-10,-459,-164087,8681,100422,76212,-5378802,
%T -39034164,179672159,4829750225,6386525014044,-433950503293,
%U -5980728715903,10978530244761,574256516189624094,8612076567202811
%N arccosh(exp(x)-sin(x))=x+6/2!*x^2+259/6!*x^6+19/7!*x^7-10/8!*x^8...
%K sign
%O 0,2
%A Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)

We replace this with two sequences:
Calculating in Maple, here is what I get:

t1 := arccosh(exp(x) - sin(x));
t2:=series(%,x,50);

               2         3         4          5    503   6    521    7    13    8
t2 := x + 1/3 x  - 1/18 x  - 1/27 x  - 5/648 x  + ----- x  + ------ x  - ----- x  -
                                                  68040      136080      51030

     827    9    1812469    10     44303839    11     9150959    12     17949817     13
    ------ x  - ---------- x   + ------------ x   + ----------- x   + ------------- x
    653184      4849891200       203695430400       43649020800       1466607098880

...

or

1*x+1/3*x^2-1/18*x^3-1/27*x^4-5/648*x^5+503/68040*x^6+521/136080*x^7-13/51030*x^8...

Extract the coefficients:


> t3:=seriestolist(t2);
                  -1  -1  -5    503    521     -13    -827    -1812469     44303839
t3 := [0, 1, 1/3, --, --, ---, -----, ------, -----, ------, ----------, ------------,
                  18  27  648  68040  136080  51030  653184  4849891200  203695430400

      9150959      17949817      -17645163013     -256103154817      128934425819
    -----------, -------------, ---------------, ----------------, -----------------,
    43649020800  1466607098880  285988384281600  8579651528448000  15014390174784000

...

> t4:=map(numer,t3);
t4 := [0, 1, 1, -1, -1, -5, 503, 521, -13, -827, -1812469, 44303839, 9150959, 17949817,

    -17645163013, -256103154817, 128934425819, 869866425977, 7718787745279031,

    -657691477236554437, -23770118746323659, 9983376979312685189, 15480320293784856319969,

    240884767523727873137, -3694316464734123751573, -12325434071961822974423,

> t5:=map(denom,t3);

1, 1, 3, 18, 27, 648, 68040, 136080, 51030, 653184, 4849891200, 203695430400,

    43649020800, 1466607098880, 285988384281600, 8579651528448000, 15014390174784000,

    64061398079078400, 2793397263238213632000, 184364219373722099712000,

    9669452065055354880000, 46459783282177969127424000, 16028625232351399348961280000,

and these will be the new A013305 and its new companion A202359.
The definitions will be:
Numerators of coefficients in Taylor series expansion of arccosh(exp(x)-sin(x))
Denominators of ditto
and they get keyword frac and cross-reference each other,
and we put
x+1/3*x^2-1/18*x^3-1/27*x^4-5/648*x^5+503/68040*x^6+521/136080*x^7-13/51030*x^8...
in the example line for both of them.

I suppose one could make the argument that since the wrong values
have been in the OEIS for 15 years, we should leave them and mark them as
"dead", and say "Incorrect sequence based on erroneous calculation of 
A202359/A202360, and have TWO new sequences for the correct values.

What do people think?

I was going to edit A013305 as an example, but I will wait.

The question is, do we keep the rotten sequence, marked "dead",
and create 2 new sequences,
or replace it with the numerators and have 1 new sequence for the denominators?

Neil