\\% Basic manipulations of vectors.
\\ Author: Ralf Stephan, (reformatted by Joerg Arndt)
\\ License: GPL version 3 or later
\\ online at http://www.jjj.de/pari/
\\ version: 2011-April-08 (19:18)
\\ gp/pari code for Doron's challenge at http://mathoverflow.net/questions/71092/how-many-integer-partitions-of-a-googol-10100-into-at-most-60-parts/71132#71132
\\Needs http://www.jjj.de/pari/fastrec.gpi
ggf(v)=
{ /* Guess generating function for supplied vector of integers */
/* Example: ggf(vector(10,j,fibonacci(j)^2)) ==> (-x + 1)/(x^3 - 2*x^2 - 2*x + 1) */
local(l,m,p,q,B);
l = length(v);
B = floor(l/2);
if ( B<4, return(0) );
m = matrix(B, B, x, y, v[x-y+B+1]);
q = qflll(m, 4)[1];
if ( length(q)==0, return(0) );
\\ p = sum(k=1, B, x^(k-1)*q[k,1]);
p = sum(k=1, B, 'x^(k-1)*q[k,1]);
\\ q = Pol( Pol( vector(l, n, v[l-n+1]) )*p + O(x^(B+1)) );
q = Pol( Pol( vector(l, n, v[l-n+1]) )*p + O('x^(B+1)) );
if ( polcoeff(p,0)<0, q=-q; p=-p);
q = q/p;
\\ p = Ser(q+O(x^(l+1)));
p = Ser(q+O('x^(l+1)));
for (m=1, l, if( polcoeff(p,m-1)!=v[m], return(0)) );
return(q);
} /* ------- */
\\ aux:
strsgn(a)=if(a>=0, Str("+" a), Str(a));
\\prtpol(c)=
\\{ /* print coefficients with ascending degrees */
\\ print1("(");
\\ for (d=(poldegree(c)-...)
\\ print1(")");
\\} /* ----- */
\\
\\prtgf(v)=
\\{ /* Print Generating function */
\\ local(gf, t);
\\ gf = ggf(v);
\\ if ( gf==0, return(0) );
\\ t = numerator(gf);
\\ print1
\\ return(1);
\\} /* ----- */
prtrec(v)=
{ /* Print recurrence relation */
local(c,d,c0,t);
c=ggf(v);
if ( c==0, return(0) );
c = denominator(c);
\\ print(c);
\\ c = polrecip(c);
\\ print(c);
d = poldegree(c);
c0 = polcoeff(c,0);
print1("a(n) =");
for (k=1, d,
t=polcoeff(c,k)/c0;
if (t, print1(" "); print1( strsgn( -t ), "*a(n-",k,")" ) );
);
print();
return(1);
} /* ----- */
prtrec2(c)=
{ /* Print recurrence relation */
local(d,c0,t,r,t2);
\\c=ggf(v);
if ( c==0, return(0) );
c = denominator(c);
\\ print(c);
\\ c = polrecip(c);
\\ print(c);
d = poldegree(c);
r=[];\\listcreate(d);
c0 = polcoeff(c,0);
print1("a(n) =");
forstep (k=1, d,1,
t=polcoeff(c,k)/c0;
t2=-polcoeff(c,k)/c0;
r=concat(r,t2);
if (t, print1(" "); print1( strsgn( -t ), "*aa(n-",k,")" ) );
);
print();
return(r);
} /* ----- */
k4=60;
d=prod(n=1,k4,(1-x^n));
{aa(n)=return(polcoeff(1/d+O(x^(n+1)),n ));};
zz=prtrec2(1/d);
a0=[];\\listcreate(poldegree(d));
for(n=0,poldegree(d)-1,a0=concat(a0,aa(n)); );
{getd()=d;}
{geta0()=a0;}
{getzz()=zz;}
X=frec(a0,zz,10^100)[1];
\\ ==== end of file ====