A000487 Number of permutations of length n with exactly two valleys. (Formerly M5022 N2165)

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

Offset

5,1

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).

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

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 *)

Crossrefs

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

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Ralf Stephan, Sep 18 2003

Status

approved