correct me if i ma wrong this is a method i cooked up

 

let S = c1 + c5+ c9 ........

 

(1+x) ⁿ = c0 + c1x + c2 x^{2 + .......}

 

let @ = 1^1/4

(1+@)ⁿ  = c0 + c1@ +c2 @²  + ....

multiply both sides by @³

@³ (1+@)ⁿ  = S + @ ( c2 + c6+ c10 + ...) + @² (c3+ c7+ c11+ ...) +@³ ( c0 + c4 + c8 + ..)   -------------------(1)

 

in the above equation

substitute the  4 values for @ ie the four roots of the equation @ = 1^1/4

 giving you 4 different equations and all 4

and u will be left with S on RHS ( 1 + @ + @² + @³ =0) and some value on LHS

 

and this method can be generalised for all AP in 'r' for nCr

 

rate me if u find it useful and correct me if i am wrong