login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A000517
Number of permutations of length n with exactly three valleys.
(Formerly M5431 N2360)
6
272, 7936, 137216, 1841152, 21253376, 222398464, 2174832640, 20261765120, 182172651520, 1594922762240, 13684856848384, 115620218667008, 965271355195392, 7984436548730880, 65569731961159680, 535438370914959360, 4353038473793372160, 35266789418949672960
OFFSET
7,1
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
Désiré André, Mémoire sur les séquences des permutations circulaires, Bulletin de la S. M. F., tome 23 (1895), pp. 122-184.
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 (40, -700, 7056, -45360, 194304, -561728, 1082624, -1332224, 946176, -294912).
FORMULA
G.f.: 16x^7(17-184x+636x^2-720x^3)/((1-2x)^4*(1-4x)^3*(1-6x)^2*(1-8x)). - Ralf Stephan, Sep 18 2003 [Proved by Désiré André, 1895, p.154, for circular permutations (see A008303). Peter Luschny, Aug 07 2019]
MATHEMATICA
nn = 20; Drop[CoefficientList[Series[16 x^7 (17 - 184 x + 636 x^2 - 720 x^3)/((1 - 2 x)^4*(1 - 4 x)^3*(1 - 6 x)^2*(1 - 8 x)), {x, 0, nn}], x], 7] (* T. D. Noe, Jun 20 2012 *)
LinearRecurrence[{40, -700, 7056, -45360, 194304, -561728, 1082624, -1332224, 946176, -294912}, {272, 7936, 137216, 1841152, 21253376, 222398464, 2174832640, 20261765120, 182172651520, 1594922762240}, 20] (* Jean-François Alcover, Feb 09 2016 *)
CROSSREFS
Column k=3 of A008303.
Sequence in context: A316930 A317701 A141546 * A350975 A250341 A168372
KEYWORD
nonn
EXTENSIONS
More terms from Ralf Stephan, Sep 18 2003
STATUS
approved