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A000487
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Number of permutations of length n with exactly two valleys.
(Formerly M5022 N2165)
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6
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16, 272, 2880, 24576, 185856, 1304832, 8728576, 56520704, 357888000, 2230947840, 13754155008, 84134068224, 511780323328, 3100738912256, 18733264797696, 112949304754176, 680032201605120, 4090088616099840, 24582312700149760, 147669797096652800
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OFFSET
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5,1
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REFERENCES
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F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 5..200
Désiré André, Mémoire sur les séquences des permutations circulaires, Bulletin de la S. M. F., tome 23 (1895), pp. 122-184.
Nelson H. F. Beebe, The Greek functions: gamma, psi, and zeta, In: The Mathematical-Function Computation Handbook, 2017. See pp. 549-550.
C. J. Fewster, D. Siemssen, Enumerating Permutations by their Run Structure, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
R. G. Rieper and M. Zeleke, Valleyless Sequences, arXiv:math/0005180 [math.CO], 2000.
Index entries for linear recurrences with constant coefficients, signature (20,-160,656,-1456,1664,-768).
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FORMULA
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G.f.: 16x^5(1-3x)/((1-2x)^3*(1-4x)^2*(1-6x)). - Ralf Stephan, Sep 18 2003 [Proved by Désiré André, 1895, p. 154, for circular permutations (see A008303). Peter Luschny, Aug 07 2019]
a(n) = (6^n + (2 - 2n)4^n + (2n^2 - 4n - 1)2^n)/32. - Mitchell Harris, Apr 02 2004
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MATHEMATICA
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nn = 30; Drop[CoefficientList[Series[16 x^5 (1 - 3 x)/((1 - 2 x)^3*(1 - 4 x)^2*(1 - 6 x)), {x, 0, nn}], x], 5] (* T. D. Noe, Jun 20 2012 *)
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CROSSREFS
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Cf. A000431, A000517, A130651.
Column k=2 of A008303.
Sequence in context: A144660 A158574 A330151 * A249391 A197622 A002303
Adjacent sequences: A000484 A000485 A000486 * A000488 A000489 A000490
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Ralf Stephan, Sep 18 2003
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STATUS
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approved
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