Some Properties Of Catalan-Larcombe-French Numbers

Let {Pn} be the Catalan-Larcombe-French numbers given by P0=1,P1?8 and n2Pn = 8(3n2—3n+1)Pn-1-128(n—1)2Pn-2(n?2),and let Sn=Pn/2n.In this thesis we use Lucas theorem and its extension to deduce congruences for Snp,Snp+1(mod p3),Smpr-1(mod pr)and Smpr?1(mod p2r),where p is an odd prime and m,n,r are positive integers.We also use mathematical induction to prove that S(p2-1)/2?O(mod p2)for any prime p?5,7(mod 8),and(1+(1/m(m-1))Sm2>Sm+1Sm-1 for m=2,3,....