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%I M5022 N2165 #35 Feb 01 2022 00:59:21
%S 16,272,2880,24576,185856,1304832,8728576,56520704,357888000,
%T 2230947840,13754155008,84134068224,511780323328,3100738912256,
%U 18733264797696,112949304754176,680032201605120,4090088616099840,24582312700149760,147669797096652800
%N Number of permutations of length n with exactly two valleys.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000487/b000487.txt">Table of n, a(n) for n = 5..200</a>
%H Désiré André, <a href="https://doi.org/10.24033/bsmf.519">Mémoire sur les séquences des permutations circulaires</a>, Bulletin de la S. M. F., tome 23 (1895), pp. 122-184.
%H Nelson H. F. Beebe, <a href="https://dx.doi.org/10.1007/978-3-319-64110-2_18">The Greek functions: gamma, psi, and zeta</a>, In: The Mathematical-Function Computation Handbook, 2017. See pp. 549-550.
%H C. J. Fewster, D. Siemssen, <a href="http://arxiv.org/abs/1403.1723">Enumerating Permutations by their Run Structure</a>, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
%H R. G. Rieper and M. Zeleke, <a href="https://arxiv.org/abs/math/0005180">Valleyless Sequences</a>, arXiv:math/0005180 [math.CO], 2000.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (20,-160,656,-1456,1664,-768).
%F 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]
%F a(n) = (6^n + (2 - 2n)4^n + (2n^2 - 4n - 1)2^n)/32. - Mitchell Harris, Apr 02 2004
%t 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 *)
%Y Cf. A000431, A000517, A130651.
%Y Column k=2 of A008303.
%K nonn,easy
%O 5,1
%A _N. J. A. Sloane_
%E More terms from _Ralf Stephan_, Sep 18 2003
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