[seqfan] Re: A000041, the partitions : the classical example of p(200), macmahon, hardy and ramanujan
Simon Plouffe
simon.plouffe at gmail.com
Tue Mar 15 05:05:58 CET 2011
Yes it is!,
the expression with the GAMMA function
and power of pi is not the classical formula
at all, it resembles only.
I made this exercise to see if it was possible
to get the famous p(200) computation without
using all the theory behind it and just the plain
decimal expansion of 1 constant, it worked.
Actually, the number C*exp(Pi*10)^200, where
C is the constant contains the 205 coefficients
of the partitions, that is p(1) to p(205) all
at once. Once computed (with 2800 digits) it
is only a matter of extracting the coefficients
using the ordinary change of base formula which
is y(n) = [k*x(n)] and x(n+1) = {k*x(n)}, where
[ ] and { } are the floor and fractional part
of a real number. y(n) are the coefficients.
The value of p(200) is obtained at the 199'th
evaluation of the formula above.
The point is (if I may)that within the decimal
expansion of certains numbers lies the coefficients
of some sequences and vice-versa, take this example :
when f(x) = 1/sqrt(1-4*x), evaluated at x=1/100
will give the first few coefficients of binomial(2*n,n).
The coefficients can be <seen> directly since
they occur in this case in base 100. I wanted to
see if that idea could be applied to more
interesting examples. Since the base is exp(Pi*10)
then each term is separated by something like 14
decimal digits, exp(Pi*10) is about 4.4x10^13.
This is why the computation of p(200) needs 2800
digits to be valid.
The example of 1/sqrt(1-4*x) is obvious the
case of p(n) is less obvious but equally valid and
much more interesting!,
I could get the value of F(16), it is an horrible
expression that contains an algebraic number of
degree 32, Mathematica was quite excellent to
extract the algebraic number and Mr Gosper
got the value of F(24) which
could also be used to get more coefficients. The
big drawback is of course the numerical precision
needed to complete the calculation. This is
an academic exercise but I find it very fruitful.
The formula for F(32) is an algebraic number of
degree 64, I don't think it is elegant.
best regards,
simon plouffe
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