Subsequent factor of binomial coefficients

by fungarwai, Sep 20, 2020, 11:31 AM

This post has been edited 7 times. Last edited by fungarwai, Mar 20, 2021, 12:52 PM

Chromatic polynomial of a 3×n grid

by fungarwai, Apr 26, 2020, 1:26 AM

Denote Chromatic polynomial of a m×n grid as $P(G_{m,n},t)$

Chromatic polynomial of a 2×n grid has been found and named as ladder graph

$P(G_{2,n},t)=t(t-1)(t^2-3t+3)^{n-1}$

My recurrence equation found for Chromatic polynomial of a 3×n grid is as follow

$P(G_{3,n},t)=(t^3-5t^2+11t-10)P(G_{3,n-1},t)-(t^4-7t^3+19t^2-24t+11)P(G_{3,n-2},t)$
$~=(t-2)(t^2-3t+5)P(G_{3,n-1},t)-(t-1)(t^3-6t^2+13t-11)P(G_{3,n-2},t)$

$P(G_{3,1},t)=t(t-1)^2,~P(G_{3,2},t)=t(t-1)(t^2-3t+3)^2$
$P(G_{3,3},t)=(t^3-5t^2+11t-10)t(t-1)(t^2-3t+3)^2
-(t^4-7t^3+19t^2-24t+11)t(t-1)^2$
$~=t(t-1)(t^7-11t^6+55t^5-161t^4+298t^3-350t^2+244t-79)$

[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+7; i=i+1){
if(i!=10){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
draw((k+6,1)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,1));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (31,1));
k=32;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=[(t-1)^3-(t-2)^2]$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-4t^2+7t-5)$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy][asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,2)--(k+6,0));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy][asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-7,1));label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (23,1));
k=24;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)]$", (3,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (15,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-11,1));label("$=(t-1)(t-2)$", (3,1));
k=7;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (14,1));
k=15;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=[(t-1)^3-(t-2)^2]$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=[(t-1)^3-(t-2)^2-(t-1)^2+(t-1)]$", (11,1));
k=20;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (26,1));
k=27;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-5t^2+10t-7)$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
dot((k+6,1));label("...", (k+2,0.5));label("...", (k+2,1.5));label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (39,1));
k=40;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]


[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)+1]$", (3,1));
k=8;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (14,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=(t^2-3t+3)$", (3,1));
k=6;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (12,1));
k=14;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
$a_n=(t^3-5t^2+10t-7)a_{n-1}-b_{n-1}$
$b_n=a_n-(t^2-3t+3)a_{n-1}+(t-2)b_{n-1}$

$b_{n-1}=(t^3-5t^2+10t-7)a_{n-1}-a_n$
$b_n=(t^3-5t^2+10t-7)a_n-a_{n+1}$
$(t^3-5t^2+10t-7)a_n-a_{n+1}=a_n-(t^2-3t+3)a_{n-1}+(t-2)[(t^3-5t^2+10t-7)a_{n-1}-a_n]$
$a_{n+1}=(t^3-5t^2+11t-10)a_n-(t^4-7t^3+19t^2-24t+11)a_{n-1}$

[asy]
unitsize(5mm);int i, j, k;
label("$a_1:$", (0,1));
k=2;
for(i=k; i<k+1; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){dot((i,j));}}
label("$a_2:$", (5,1));
k=7;
for(i=k; i<k+2; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+1,j));
dot((i,j));}}
label("$a_3:$", (11,1));
k=13;
for(i=k; i<k+3; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+2,j));
dot((i,j));}}
[/asy]

$a_1=t(t-1)^2,~a_2=t(t-1)(t^2-3t+3)^2$
$a_3=(t^3-5t^2+11t-10)t(t-1)(t^2-3t+3)^2-(t^4-7t^3+19t^2-24t+11)t(t-1)^2$
$~=t(t-1)(t^7+11t^6+55t^5-161t^4+298t^3-350t^2+244t-79)$
This post has been edited 17 times. Last edited by fungarwai, Aug 27, 2021, 12:50 PM

The module distribution for Pascal's triangle row

by fungarwai, Sep 13, 2019, 12:14 PM

Let $n=\sum_{k=0}^\infty n_k p^k, N_r=\{k|n_k=r\}$

For p=2,

There are $2^{|N_1|}$ different ways to choose $m$ for $\binom{n}{m}\equiv 1\pmod{2}$

For p=3,

There are $2^{|N_1|}\cdot \frac{3^{|N_2|}-1}{2}$ different ways to choose $m$ for $\binom{n}{m}\equiv 2\pmod{3}$ and
$2^{|N_1|}\cdot \frac{3^{|N_2|}+1}{2}$ different ways to choose $m$ for $\binom{n}{m}\equiv 1\pmod{3}$

Proof

For p=5,

The number of $\binom{n}{m}\equiv 1\pmod{5}$ is $
\frac{1}{4}2^{|N_1|}5^{|N_4|}(3^{|N_2|}4^{|N_3|}+0^{|N_3|})
+\frac{1}{2}2^{|N_1|}5^{\frac{|N_2|}{2}}2^{\frac{3|N_3|}{2}}
\cos\left(|N_2|\tan^{-1}\frac{1}{2}-|N_3|\frac{\pi}{4}\right)$
The number of $\binom{n}{m}\equiv 2\pmod{5}$ is $
\frac{1}{4}2^{|N_1|}5^{|N_4|}(3^{|N_2|}4^{|N_3|}-0^{|N_3|})
+\frac{1}{2}2^{|N_1|}5^{\frac{|N_2|}{2}}2^{\frac{3|N_3|}{2}}
\sin\left(|N_2|\tan^{-1}\frac{1}{2}-|N_3|\frac{\pi}{4}\right)$
The number of $\binom{n}{m}\equiv 3\pmod{5}$ is $
\frac{1}{4}2^{|N_1|}5^{|N_4|}(3^{|N_2|}4^{|N_3|}-0^{|N_3|})
-\frac{1}{2}2^{|N_1|}5^{\frac{|N_2|}{2}}2^{\frac{3|N_3|}{2}}
\sin\left(|N_2|\tan^{-1}\frac{1}{2}-|N_3|\frac{\pi}{4}\right)$
The number of $\binom{n}{m}\equiv 4\pmod{5}$ is $
\frac{1}{4}2^{|N_1|}5^{|N_4|}(3^{|N_2|}4^{|N_3|}+0^{|N_3|})
-\frac{1}{2}2^{|N_1|}5^{\frac{|N_2|}{2}}2^{\frac{3|N_3|}{2}}
\cos\left(|N_2|\tan^{-1}\frac{1}{2}-|N_3|\frac{\pi}{4}\right)$

Proof

Example
This post has been edited 1 time. Last edited by fungarwai, Sep 13, 2019, 12:15 PM

Notable algebra methods with proofs and examples

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  • Nice blog!

    by Inconsistent, Mar 18, 2024, 2:41 PM

  • hey, nice blog! really enjoyed the content here and thank you for this contribution to aops. Sure to subscribe! :)

    by thedodecagon, Jan 22, 2022, 1:33 AM

  • thanks for this

    by jasperE3, Dec 3, 2021, 10:01 PM

  • I am working as accountant and studying as ACCA student now.
    I graduated applied mathematics at bachelor degree in Jinan University but I still have no idea to find a specific job with this..

    by fungarwai, Aug 28, 2021, 4:54 AM

  • Awesome algebra blog :)

    by Euler1728, Mar 22, 2021, 5:37 AM

  • I wonder if accountants need that kind of math tho

    by Hamroldt, Jan 14, 2021, 10:55 AM

  • Nice!!!!

    by Delta0001, Dec 12, 2020, 10:20 AM

  • this is very interesting i really appericate it :)

    by vsamc, Oct 29, 2020, 4:42 PM

  • this is god level

    by Hamroldt, Sep 4, 2020, 7:48 AM

  • Super Blog! You are Pr0! :)

    by Functional_equation, Aug 23, 2020, 7:43 AM

  • Great blog!

    by freeman66, May 31, 2020, 5:40 AM

  • cool thx! :D

    by erincutin, May 18, 2020, 4:55 PM

  • How so op???

    by Williamgolly, Apr 30, 2020, 2:42 PM

  • Beautiful

    by Al3jandro0000, Apr 25, 2020, 3:11 AM

  • Nice method :)

    by Feridimo, Jan 23, 2020, 5:05 PM

  • This is nice!

    by mufree, May 26, 2019, 6:40 AM

  • Wow! So much Algebra.

    by AnArtist, Mar 15, 2019, 1:19 PM

  • :omighty: :omighty:

    by AlastorMoody, Feb 9, 2019, 5:17 PM

  • 31415926535897932384626433832795

    by lkarhat, Dec 25, 2018, 11:53 PM

  • rip 0 shouts and 0 comments until now

    by harry1234, Nov 17, 2018, 8:56 PM

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