Identification of  Factors: the Andrews - Burge Determinant

m[i_,j_]:=Binomial[mu+i+j,2i-j];

Rows[n_,val_]:=(
Var=Sum[c[i]*Table[m[i,j]/.mu->val,{j,0,n-1}],{i,0,n-1}];
Var=Solve[Var==Table[0,{n}],Table[c[i],{i,0,n-1}]];
Table[c[i],{i,0,n-1}]/.Var);
Columns[n_,val_]:=(
Var=Sum[c[j]*Table[m[i,j]/.mu->val,{i,0,n-1}],{j,0,n-1}];Var=Solve[Var==Table[0,{n}],Table[c[j],{j,0,n-1}]];
Table[c[j],{j,0,n-1}]/.Var);

Rows[4,-4]

<<zb.m

Zb[Binomial[n,k],{k,0,n},n,1]

Zb[Binomial[n-2,j-1]Binomial[-n+i+j,2i-j],{j,0,n-1},n,2]

LU - Factorisation: the Vandermonde Determinant (again)

U[n_]:=Table[Switch[Sign[i-j],-1,g[i,j],0,1,1,0],{i,0,n-1},{j,0,n-1}]
L[n_]:=Table[If[i≥j,k[i,j],0],{i,0,n-1},{j,0,n-1}]

LU1[n_]:=Module[{Var},Var=Solve[Map[(#==0)&,Flatten[Transpose[W[n]].U[n]-L[n]]],Flatten[Join[Table[k[i,j],{i,0,n-1},{j,0,i}],Table[g[i,j],{i,0,n-1},{j,i+1,n-1}]]]];
UU=Factor[U[n]/.Var[[1]]];LL=Factor[L[n]/.Var[[1]]];
Print[""];Print["Your upper triangular matrix is"];Print[""];
Print[UU];
Print[""];Print["Your lower triangular matrix is"];Print[""];
Print[LL];];
LU2[n_]:=Module[{Var},Var=Solve[Map[(#==0)&,Flatten[W[n].U[n]-L[n]]],Flatten[Join[Table[k[i,j],{i,0,n-1},{j,0,i}],Table[g[i,j],{i,0,n-1},{j,i+1,n-1}]]]];
UU=Factor[U[n]/.Var[[1]]];LL=Factor[L[n]/.Var[[1]]];
Print[""];Print["Your upper triangular matrix is"];Print[""];
Print[UU];
Print[""];Print["Your lower triangular matrix is"];Print[""];
Print[LL];]

W[n_]:=Table[x[i]^(j),{i,0,n-1},{j,0,n-1}]

LU1[5]

Interlude: A Word about Guessing

<<rate2.m

Rate[1,3,6,10]

Rate[1,2,6,24]

Rate[1,2,5,14,42]

Rate[1,2,7,42,429,7436,218348,10850216]

Rate[2,16,84,400,1820,8064,35112]

Rate[1,1,2,3,5,8,13]

<<RISC`Guess`

GuessMinRE[{1,1,2,3,5,8,13,21,34},f[n]]

GuessMinRE[{1,2,6,24,120,720},f[n]]

GuessMinRE[{1,1,2,4,9,21,51,127,323,835,2188,5798,15511,41835},f[n]]

LU - Factorisation: the Mills-Robbins-Rumsey Determinant (again)

W[n_]:=Table[FunctionExpand[Binomial[mu+i+j,2i-j]],{i,0,n-1},{j,0,n-1}];

LU2[12]

<<rate2.m

Table[UU[[2,i+1]],{i,1,11}]

Apply[Ratekurz,%]

Method 5: The holonomic Ansatz

a[i_,j_]:=Binomial[i+j,j];
Clear[c];c[n_,k_]:=0/;n<k;
Module[{Var,cVar},
For[n=1,n<21,n++,
c[n,n]=1;cVar=Table[c[n,k],{k,1,n-1}];Var=Solve[Table[Sum[a[l,k]c[n,k],{k,1,n}]==0,{l,1,n-1}],cVar];
For[k=1,k<n,k++,c[n,k]=Var[[1,k,2]]];
]
]

data=Table[c[n,k],{n,1,10},{k,1,10}]

TableForm[%]

Clear[n,k];GuessMultRE[data,{f[n,k],f[n+1,k],f[n+1,k+1],f[n+1,k+2],f[n,k+1],f[n,k+2]},{n,k},1]

<<RISC`HolonomicFunctions`

ann1=Annihilator[a[i,k],{S[k],S[n]}]

{-(2+k) f[n,1+k]+(3+k+n) f[n,2+k]+(1+n) f[1+n,2+k],(2+2 k-n) f[n,k]-3 (2+k) f[n,1+k]+(3+k) f[n,2+k]-(2+k) f[1+n,1+k]+(3+k) f[1+n,2+k],-(1+k) f[n,k]+(2+k+n) f[n,1+k]+(1+n) f[1+n,1+k]}

{-(2+k)S[k]+(3+k+n) S[k]^2+(1+n) S[n]S[k]^2,(2+2 k-n)-3 (2+k)S[k]+(3+k)S[k]^2-(2+k) S[n]S[k]+(3+k) S[n]S[k]^2,-(1+k) +(2+k+n)S[k]+(1+n) S[n]S[k]}

ToOrePolynomial[%]

ann2=OreGroebnerBasis[%]

DFiniteTimes[ann1,ann2]

CreativeTelescoping[%,S[k]-1,{S[n]}]

A Hankel Determinant of Bernoulli numbers

HankB[n_]:=Det[Table[BernoulliB[i+j+2],{i,0,n-1},{j,0,n-1}]]

Table[HankB[n],{n,1,10}]

Apply[Ratekurz,%]