An extension of Euler numbers and alternating permutation records

Abstract We study the sequence of polynomials C n ( x , y ) defined through the recurrence C 0 ( x , y ) = 1, C n ( x , y ) = x ( y + 1) C n −1 ( x + 2, y + 2) − xyC n −1 ( x , y ), which turns out to be an extension of Euler numbers. We give a combinatorial interpretation of these numbers in terms of down-up permutations with respect to the numbers of even and odd upper records and a continued fraction expansion for their ordinary generating function.