Symmetrical, extracellular fibrils, which are related to the "special fibrils" of the dermis described by Palade and Farquhar, have been found along the outer surface of the basement membrane covering the notochord in the tail of Rana catesbeiana (bullfrog) tadpoles. The fibrils are approximately 7,500 A long and occur singly or in clusters. The single fibrils are characterized by a symmetrical...

We give three proofs for the following symmetrical identity involving binomial coefficients ( n m ) and Eulerian numbers 〈 n m 〉 : ∑ k ( a + b k ) 〈 k a − 1 〉 = ∑ k ( a + b k ) 〈 k b − 1 〉 for any positive integers a and b (where we take 〈 0 0 〉 = 0). We also show how this fits into a family of similar (but more complicated) identities for Eulerian numbers.