The Value of Certain Combinatorics Sum
In this paper we analyze the values and the properties of the function S(n,l):=_k=0^n(-1)^knk(n-k)^l (n,lN{ 0} ), for n<l. At first, we obtain two recurrence relations. Namely, we prove that for every nN { 0} and every lN such that l>n, we have equation* S(n+1,l)=_k=1^l-nlkS(n,l-k), equation* and also, for every nN{ 0} and every lN, 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) =_k_1=1^l-(n-1)lk_1 _k_2=1^l-k_1-(n-2)l-k_1 k_2 ·_k_3=1^l-k_1-k_2-(n-3)l-k_1-k_2 k_3 … _k_n-1=1^l-_i=1^n-2k_i-1 l-_i=1^n-2k_i k_n-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 , number=2, journal=Sarajevo Journal of Mathematics, author=Vugdalić, Ramiz, year=2024, month=Jun., pages=247–255