Relation between square and centered pentagonal numbers

Let s κ(n) denote the number of representations of integer n as a sum of κ squares and c κ(n) denote the number of representations of integer n as a sum of κ centered pentagonal numbers. we derive the relation s κ (8n-3κ/5)=α κ c κ (n) where α κ=2 κ + 2 κ-1 (κ/4) for 1 ≤ κ ≤ 7. we give a conjecture on the relation between s λ (n) and c λ(n) is given by β λc λ(n) =s λ (8n-3κ/5) for all integers n and λ = (λ 1,...λ m) where β λ = 2 m+2 m-1 and 1 ≤ κ ≤ 7. a special case of this conjecture is proved in which κ= 7 and λ =(3,2,1,1).