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"An (A,A) field (A is spin) contains terms with only integer spins 2A,2A-1,...,0, and corresponds to a traceless symmetric tensor of rank 2A.(Note that the number of independent components of a symmetric tensor of rank 2A in four(space-time) dimensions is:

{4.5...(4+2A-1)}/(2A)!=(3+2A)!/6(2A)!

and the tracelessness condition reduces this to:

{(3+2A)!/6(2A)!}-{(1+2A)!/6(2A-2)!}=(2A+1)^2

as expected for an (A,A) field"

I can not derive the number of components of a tensor of rank 2A in four dimensions is

4.5...(4+2A-1) and the number of tracelessnes conditions is (1+2A)!/6(2A-2)!