A000563 Number of discordant permutations.

13, 192, 1085, 3880, 10656, 24626, 50380, 94128, 163943, 270004, 424839, 643568, 944146, 1347606, 1878302, 2564152, 3436881, 4532264, 5890369, 7555800, 9577940, 12011194, 14915232, 18355232, 22402123, 27132828, 32630507, 38984800, 46292070

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
• 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

• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (6,-15, 20,-15,6,-1).

Programs

Magma

[81/40*n^5-135/4*n^4+1719/8*n^3-2487/4*n^2+3463/5*n: n in [5..45]]; // Vincenzo Librandi, Feb 10 2016

Maple

r := n->81/40*n^5-135/4*n^4+1719/8*n^3-2487/4*n^2+3463/5*n; seq(r(n), n=5..40); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
A000563:=-(-13-114*z-128*z**2+10*z**3-6*z**4+8*z**5)/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {13, 192, 1085, 3880, 10656, 24626}, 30] (* Jean-François Alcover, Feb 10 2016 *)

Formula

From Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001: (Start)

G.f.: -x^5(8x^5 - 6x^4 + 10x^3 - 128x^2 - 114x - 13) / ((1 - x)^6).

a(n) = 81/40n^5 - 135/4n^4 + 1719/8n^3 - 2487/4n^2 + 3463/5n, n>4. (End)

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001