Hoffman, Michael Derivative polynomials, Euler polynomials, and associated integer sequences. (English) Zbl 0933.11005 Electron. J. Comb. 6, No. 1, Research paper R21, 13 p. (1999); printed version J. Comb. 6, 307-319 (1999). Author’s abstract: Let \(P_n\) and \(Q_n\) be the polynomials obtained by repeated differentiation of the tangent and secant functions, respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the \(P_n\) and \(Q_n\). For example, \(P_n(0)\) and \(Q_n(0)\) are respectively the \(n\)th tangent and secant numbers, while \(P_n(0)+Q_n(0)\) is the \(n\)th André number. The André numbers, along with the numbers \(Q_n(1)\) and \(P_n(1)-Q_n(1)\), are the Springer numbers of root systems of types \(A_n\), \(B_n\), and \(D_n\) respectively, or alternatively (following V. I. Arnol’d) count the number of “snakes” of these types. We prove this for the latter two cases using combinatorial arguments. We relate the values of \(P_n\) and \(Q_n\) at \(\sqrt 3\) to certain “generalized Euler and class numbers” of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. A. Readdy. Finally, we express the values of Euler polynomials at any rational argument in terms of \(P_n\) and \(Q_n\), and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials. Cited in 22 Documents MSC: 11B83 Special sequences and polynomials 11B68 Bernoulli and Euler numbers and polynomials 05A15 Exact enumeration problems, generating functions Keywords:tangent numbers; André numbers; Springer numbers; snakes; generalized Euler numbers; class numbers; secant numbers PDFBibTeX XMLCite \textit{M. Hoffman}, Electron. J. Comb. 6, No. 1, Research paper R21, 13 p. (1999; Zbl 0933.11005) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Euler (or secant or ”Zig”) numbers: e.g.f. (even powers only) sec(x) = 1/cos(x). Generalized Euler numbers c(3,n). Generalized Euler numbers, or Springer numbers. Glaisher’s H numbers. a(n) = A000364(n) * (3^(2*n+1) + 1)/4. Glaisher’s T numbers. Springer numbers associated with symplectic group. Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k>0 or if k<0; T(n,k) = k*T(n-1,k-1) + (k+1)*T(n-1,k+1). Triangle read by rows: coefficients in polynomials P_n(u) arising from the expansion of D^(n-1) (tan x) in increasing powers of tan x for n>=1 and 1 for n=0. P_{2n+1}(sqrt(3)) (see A155100). Numerator of Euler(n,1/3). Denominator of Euler(n,1/3). Numerator of Euler(n, 1/6). Denominator of Euler(n, 1/6). Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx.