Re: [math-fun] [seqfan] A000041, the partitions : the classical example of p(200), macmahon, hardy and ramanujan
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
participants (1)
-
Simon Plouffe