EULER NUMBERS or EULERIAN NUMBERS, are the coefficients of the expansion: They were so named by the German mathematician Scherk after the man who first discovered their significance in analysis. Al though not of such wide application as the Bernouilli num bers, they are interesting on account of their intimate connec tion with the former, and of their various theoretical properties. They are also used in the summation of certain series. The first nine of these numbers were computed by Euler (q.v.). The ninth was found to be erroneous by Rothe, who gave the correct value. H. F. Scherk (182 5) computed six more, Glaisher extended the list to 27, and S. A. Joffe computed 23 more, making 5o in all.
Euler used the recurrent formula Binet made the very interesting observation that the nth Eulerian number is equal to the number of permutations that can be formed of 2n elements, ai, . . . a2n, such that the index of any element is either larger or smaller than each of the two adjacent indices, e.g., out of the 24 permutations of the four elements, the following satisfy the above condition: totalling 5 distinct permutations, and hence E2 = 5.
Of the theoretical properties of the Eulerian numbers the fol lowing are of interest : I. Every Eulerian number is a positive odd integer.
2. The sum of any two successive Eulerian numbers is divisible by 3.
3. When n is even, (mod. 3) 4. When n is odd, — 1=0 (mod. 3) 5. always ends in 1 or 5.
The first io Euler numbers are 1, 5, 61, 1385, 5o52I, 2702765, 199360981, 19391512145, 370371188237525; and E50 is a number of 127 figures.