Andr?then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index ( also called tangent numbers ) and the alternating permutations of even size by the Euler numbers of even index ( also called secant numbers ).
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Andr?then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index ( also called tangent numbers ) and the alternating permutations of even size by the Euler numbers of even index ( also called secant numbers ).
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:: : Ah, yes, thank you-the co-efficients of these polynomials are the " Eulerian numbers " as decribed here and here . . . but not, apparently, currently in Wikipedia, because " Eulerian number " redirects to Euler number, which is " not " the same thing.
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These include primes, pseudoprimes, graph colorings, Euler numbers, continued fractions, Stirling numbers, Pythagorean triples, Ramsey theory, Lucas-Bernoulli numbers, quadratic residues, higher-order recurrence sequences, nonlinear recurrence sequences, combinatorial proofs of number-theoretic identities, Diophantine equations, special matrices and determinants, the Collatz sequence, public-key crypto functions, elliptic curves, fractal dimension, hypergeometric functions, Fibonacci polytopes, geometry, graph theory, music, and art.
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