Pascal's Triangle

Author

Eric W. Weisstein
January 11, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/PascalsTriangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PascalsTriangle.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Triangle

ColumnForm[Table[Binomial[n,k],{n,0,10},{k,0,n}],Center]

With[{n=5},MatrixExp[Table[If[i==(j+1),j+1,0],{i,0,n},{j,0,n}]]]//TraditionalForm

Binary Plot

<<Utilities`Plot`

Length[t=Flatten[Table[Binomial[n,k],{n,0,35},{k,0,n}]]]

Show[PixelPerfect[BasePlot[Take[t,255]],BlockSize->2]]

Show[PixelPerfect[BasePlot[Take[t,511]],BlockSize->1]]

Sum of Odd Elements

Constant

N[Log[3]/Log[2],20]

N[Log[2,3],20]

Identities

(({{n}, {r}}))=(({{n-1}, {r-1}}))+(({{n-1}, {r}}))

FullSimplify[Subtract@@{Binomial[n,r],Binomial[n-1,r]+Binomial[n-1,r-1]},(r|n)∈Integers&&n>1&&r>1]

Refine[FunctionExpand[%],(r|n)∈Integers&&n>0&&r>0]

Table[-Binomial[-1+n,-1+r]-Binomial[-1+n,r]+Binomial[n,r],{n,0,5},{r,0,n}]

Sums

(∑_i^n)(Binomial[i, j])==(Binomial[n+1, j+1])

Subtract@@@Table[{Sum[Binomial[i,j],{i,n}],Binomial[n+1,j+1]}/.{j->RandomInteger[{1,100}],n->RandomInteger[{0,100}]},{1000}]//Union

V5

FullSimplify[Sum[Binomial[i,j],{i,n}],j∈Integers]

FullSimplify[%/Binomial[n,j]]

V6

Sum[Binomial[i,j],{i,n}]

FullSimplify[%,j∈Integers&&j>0]

FullSimplify[%/Binomial[n,j]]

V7

Sum[Binomial[i,j],{i,n}]

FullSimplify[%,j∈Integers&&j>0]

Sum[Binomial[i,j],{i,n},Assumptions->Element[j,Integers]&&j≥0]

V7.0.1

Sum[Binomial[i,j],{i,n}]

FullSimplify[%,j∈Integers&&j>1]

FullSimplify[((Gamma[2+n])/(Gamma[2+j] Gamma[1-j+n]))-Binomial[1+n,1+j]]

Sum[Binomial[i,j],{i,n},Assumptions->Element[j,Integers]&&j≥0]

(∑_(j=0)^i)(Binomial[i, j])==(2^i)

Sum[Binomial[i,j],{j,0,i}]

(∑_(i=0)^(k-1))(2^i)==(2^k)-1

Sum[2^i,{i,0,k-1}]

(∑_k^n)(Binomial[k, n-k])==(Fibonacci[n+1])

Table[Sum[Binomial[k,n-k],{k,n}],{n,10}]

Table[Fibonacci[n+1],{n,10}]

V6

Sum[Binomial[k,n-k],{k,n}]

V7

Sum[Binomial[k,n-k],{k,n}]

(∑_k^n)(Binomial[k, n-k])

V6

s=Sum[Binomial[k,n-k],{k,0,Floor[n/2]}]

Table[s,{n,10}]

V7

s=Sum[Binomial[k,n-k],{k,0,Floor[n/2]}]

Table[s,{n,10}]

(∑_(k=0)^(Floor[n/2]))(Binomial[k, n-k])

V6

Sum[Binomial[k,n-k],{k,0,Floor[n/2]}]

FullSimplify[%,n∈Integers&&n>0]

V7

s=Sum[Binomial[k,n-k],{k,0,Floor[n/2]}]

Table[s,{n,10}]

(∑_(k=0)^(Floor[n/2]))(Binomial[n-k, k])==(Fibonacci[n+1])==(i^n) (Subscript[U, n])(-(i/2))

V4.2

Sum[Binomial[n-k,k],{k,0,Floor[n/2]}]

FullSimplify[(I^n) ChebyshevU[n,-(I/2)],n∈Integers&&n>0]

FullSimplify[FunctionExpand[(I^n) ChebyshevU[n,-(I/2)]],n∈Integers]

V5.0

Sum[Binomial[n-k,k],{k,0,Floor[n/2]}]

Table[%,{n,8}]//Simplify

Sum[Binomial[k,n-k],{k,n}]

FullSimplify[%,n∈Integers]

Sum[Binomial[k,n-k],{k,Floor[n/2]}]

FullSimplify[%,n∈Integers]

V7

Sum[Binomial[n-k,k],{k,0,Floor[n/2]}]

Terms

With[{nn=1000},
p=Table[Binomial[n,k],{n,nn},{k,n-1}]
];//Timing

With[{nn=3100},
p=Table[Binomial[n,k],{n,nn},{k,n-1}]
];//Timing

s=Split[Sort[Flatten[p]]];//Timing

Numbers of Rows containing n

Length[Union[First/@Position[p,#]]]&/@Range[2,20]//Times

Number of Times n Occurs

Take[First/@s,99]==Range[2,100]

Length/@Take[s,99]

Common terms

Through[{First,Length}[#]]&/@Select[s,Length[#]>4&]//Timing

Solutions Occur at least 6 times

Reduce[Binomial[n,m-1]==Binomial[n-1,m]&&m>0&&n>0,{m,n},Integers]//Timing

Reduce[Binomial[n,m-1]==Binomial[n-1,m],{m,n},Integers]//Timing

FullSimplify[FunctionExpand[
Binomial[n,m-1]-Binomial[n-1,m]/.{
n->Fibonacci[2k]Fibonacci[2k+1],
m->Fibonacci[2k-1]Fibonacci[2k]
}
],k>0&&k∈Integers]

FullSimplify[
Binomial[n,m-1]-Binomial[n-1,m]/.{
n->Fibonacci[2k]Fibonacci[2k+1],
m->Fibonacci[2k-1]Fibonacci[2k]
},k>0&&k∈Integers]

t=Table[{Fibonacci[2k]Fibonacci[2k+1],Fibonacci[2k-1]Fibonacci[2k]},{k,6}]

Binomial[n,m-1]-Binomial[n-1,m]/.Thread[{n,m}->#]&/@t

Table[Binomial[Fibonacci[2k]Fibonacci[2k+1],Fibonacci[2k-1]Fibonacci[2k]-1],{k,4}]