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  "body": "\n## Statement\n\nThe determination of the number *A*\u003csub\u003e*n*\u003c/sub\u003e of alternating permutations of the set {1, ..., *n*} is called *André's problem*. The numbers *A*\u003csub\u003e*n*\u003c/sub\u003e are variously known as *Euler numbers*, *zigzag numbers*, *up/down numbers*, or by some combinations of these names. The name Euler numbers in particular is sometimes used for a closely related sequence. The first few values of *A*\u003csub\u003e*n*\u003c/sub\u003e are 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, ... .\n\nThese numbers satisfy a simple recurrence, similar to that of the [[Catalan numbers|0YWZJEABB2]]: by splitting the set of alternating permutations (both down-up and up-down) of the set { 1, 2, 3, ..., *n*, *n* + 1 } according to the position *k* of the largest entry $n + 1$, one can show that\n\n$$2A_{n+1} = \\sum_{k=0}^n \\binom{n}{k} A_k A_{n-k}$$\n\nfor all $n \\ge 1$. (André 1881) used this recurrence to give a [[differential equation|003R0V5M7V]] satisfied by the exponential generating function\n\n$$A(x) = \\sum_{n=0}^\\infty A_n \\frac{x^n}{n!}$$\n\nfor the sequence $A_{n}$. In fact, the recurrence gives:\n\n$$2\\sum_{n\\geq 1} A_{n+1} \\frac{x^{n+1}}{(n+1)!} = \\sum_{n\\geq 1} \\sum_{k=0}^n \\frac{A_k}{k!} \\frac{A_{n-k}}{(n-k)!} \\frac{x^{n+1}}{n+1} = \\int \\left(\\sum_{k\\geq 0}A_k \\frac{x^k}{k!}\\right) \\left(\\sum_{j\\geq 0}A_j \\frac{x^j}{j!}\\right) \\, dx - x$$\n\nwhere we substitute $j = n-k$ and $\\frac{x^{n+1}}{n+1}=\\int x^{k+j}\\,dx$. This gives the integral equation\n\n$$2(A(x) - 1 - x) = \\int A(x)^2\\,dx - x,$$\n\nwhich after differentiation becomes $2\\frac{dA}{dx} - 2 = A^2-1$. This differential equation can be solved by separation of variables (using the initial condition $A(0)=A_0/0!=1$), and simplified using a tangent half-angle formula, giving the final result\n\n$$A(x) = \\tan \\left(\\frac\\pi4 + \\frac x2\\right) = \\sec x + \\tan x,$$\n\nthe sum of the secant and tangent functions. This result is known as *André's theorem*. A geometric interpretation of this result can be given using a generalization of a theorem by Johann Bernoulli.\n\nIt follows from André's theorem that the [[radius of convergence|0N288P2TB5]] of the series $A(x)$ is $\\pi$/2. This allows one to compute the asymptotic expansion\n\n$$A_n \\sim 2 \\left(\\frac{2}{\\pi}\\right)^{n + 1} n!\\,.$$\n",
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