Taylor polynomial. System of equations for the coefficients.

[marraco]

This PariGP code generates the factors of X, exactly as written here:

Code:

\\ exponent of X for which the factor will be generated.
expX=4

\\for each partition p, index i of a_i
\\vector containing vectors.
indexs=partitions(expX)
num_partitions=length(indexs)

\\exponent of lna
lna_exp=vector(num_partitions,i,length(indexs[i]))


\\exponent, in each partition, for each index of a_i
\\Row=partition
\\Column=exponent for ai
index_exp=matrix(num_partitions,expX,r,c,sum(n=1,lna_exp[r],if(c-indexs[r][n],0,1)))

\\integer divisor for each partition
divisor=vector(num_partitions,p,truncate(prod(i=1,expX,factorial(index_exp[p,i]))))

ai=[1,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12]

\\complete factor for the exponent X^expX
Factor_for_Xp=sum(p=1,num_partitions,(a*lna^lna_exp[p])/divisor[p]*prod(i=1,expX,ai[i+1]^index_exp[p,i]))

for example, for expX=4, it generates this output:

1/24*lna^4*a*a1^4 + 1/2*lna^3*a*a2*a1^2 + lna^2*a*a3*a1 + (1/2*lna^2*a*a2^2 + lna*a*a4)

This probably is not appropriate for solving it with PariGP. I don't even know if PariGP is the appropriate tool for solving the equations.

by the way, there is a procedure to programmatically generate this vector, for an arbitrary length?

ai=[1,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12]