The Value of Certain Combinatorics Sum

In this paper we analyze the values and the properties of the function S(n,l):=∑ₖ₌₀ⁿ(-1)ᵏnk(n-k)ˡ\ \(n,l∈ N∪ \ 0\ ), for $n<l.$ At first, we obtain two recurrence relations. Namely, we prove that forevery n∈ N∪ \ 0\ and every l∈ N such that $l>n,$ we have{equation*}S(n+1,l)=∑ₖ₌₁ˡ⁻ⁿ{l}{k}S(n,l-k),{equation*}and also, for every n∈ N∪ \ 0 every l∈ N, we have{equation*}S(n+1,l)=(n+1)S(n,l-1)+(n+1)S(n+1,l-1).{equation*}Further, we conclude that for every n≥ 2 and every l≥ n the following representation formula holds{multline*}S(n,l) =∑_{k₁=1}ˡ⁻⁽ⁿ⁻¹⁾{l}{k₁}∑_{k₂=1}^{l-k₁-(n-2)}{l-k₁}{k₂}\\·∑_{k₃=1}^{l-k₁-k₂-(n-3)}{l-k₁-k₂}{k₃}∑_{kₙ₋₁=1}^{l-∑ᵢ₌₁ⁿ⁻²kᵢ-1}{l-∑ᵢ₌₁ⁿ⁻²kᵢ}{kₙ₋₁}.{multline*}We obtain an explicit formula for the calculation $S(n,l),$ especially for l=n+1,,n+5, and later we give a general result. 2000 Mathematics Subject Classification. 40B05, 11Y55, 05A10