Randrianarivony, Arthur; Zeng, Jiang On an extension of Euler numbers and records of alternating permutations. (Sur une extension des nombres d’Euler et les records des permutations alternantes.) (French. English summary) Zbl 0978.05502 Sémin. Lothar. Comb. 30, B30f, 14 p. (1993). Summary: 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.The paper has been finally published under the same title in [J. Comb. Theory, Ser. A 68, No. 1, 86-99 (1994; Zbl 0809.05002)]. MSC: 05A05 Permutations, words, matrices 05A15 Exact enumeration problems, generating functions 11J70 Continued fractions and generalizations Keywords:Euler numbers; permutations; continued fraction expansion; generating function Citations:Zbl 0809.05002 PDFBibTeX XMLCite \textit{A. Randrianarivony} and \textit{J. Zeng}, Sémin. Lothar. Comb. 30, B30f, 14 p. (1993; Zbl 0978.05502) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).