A000487 Number of permutations of length n with exactly two valleys.

16, 272, 2880, 24576, 185856, 1304832, 8728576, 56520704, 357888000, 2230947840, 13754155008, 84134068224, 511780323328, 3100738912256, 18733264797696, 112949304754176, 680032201605120, 4090088616099840, 24582312700149760, 147669797096652800

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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).

Links

• 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).

Crossrefs

Cf. A000431, A000517, A130651.

Column k=2 of A008303.

Programs

Mathematica

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 *)

Formula

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

Extensions

More terms from Ralf Stephan, Sep 18 2003