Two Binomial Formulas

Generating functions

formula=(1/(2Pi))Integrate[Exp[-I q d](2 Cos[q])^10,{q,-Pi,Pi}]

formula2=formula/.{d->n-2k};
plot1=Plot@@{formula2,{k,0,n}};
plot2=DiscretePlot[Binomial[n,k],{k,0,n},PlotStyle->PointSize[Large],Filling->Bottom,Axes->None,Frame->True,PlotRange->{-50,260}];
Show[plot1,plot2,Axes->None,Frame->True]

Cauchy' s Formula

r=1;
Integrate[(((1+z)^6)/(z^3)),{z,r,r I,-r,-r I,r}]/(2Pi I)//Simplify

Binomial[6,2]

plot1=Plot3D[(((1+x+I y)^3)/((x+I y)^2))//Abs,{x,-2,2},{y,-2,2},MeshFunctions->{Max[Abs[#1],Abs[#2]]&},Mesh->1,PlotStyle->Opacity[.8],ClippingStyle->None,MeshStyle->Directive[Thick,Red]]