A000561 Number of discordant permutations.
6, 44, 145, 336, 644, 1096, 1719, 2540, 3586, 4884, 6461, 8344, 10560, 13136, 16099, 19476, 23294, 27580, 32361, 37664, 43516, 49944, 56975, 64636, 72954, 81956, 91669, 102120, 113336, 125344, 138171, 151844, 166390, 181836, 198209, 215536, 233844, 253160, 273511, 294924, 317426, 341044
Offset: 3
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
- G. C. Greubel, Table of n, a(n) for n = 3..1000
- 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 (4,-6,4,-1).
Programs
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Magma
[(9/2)*n^3-(45/2)*n^2+29*n: n in [3..45]]; // Vincenzo Librandi, Feb 10 2016
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Maple
f := n->9/2*n^3-45/2*n^2+29*n; seq(f(n), n=0..50); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001 A000561:=-(-6-20*z-5*z**2+4*z**3)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
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Mathematica
LinearRecurrence[{4, -6, 4, -1}, {6, 44, 145, 336}, 50] (* Jean-François Alcover, Feb 10 2016 *) Drop[CoefficientList[Series[x^3(6+20x+5x^2-4x^3)/(1-x)^4,{x,0,50}],x],3] (* Harvey P. Dale, Jul 20 2021 *) -
PARI
for(n=3, 45, print1(n*(9*n^2 - 45*n + 58)/2, ", ")) \\ G. C. Greubel, Nov 23 2018
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Sage
[n*(9*n^2 - 45*n + 58)/2 for n in (3..45)] # G. C. Greubel, Nov 23 2018
Formula
G.f.: x^3*(6 + 20*x + 5*x^2 - 4*x^3) / (1 - x)^4. - Jeffrey Shallit [adapted by Vincenzo Librandi, Feb 10 2016]
a(n) = n*(9*n^2 - 45*n + 58)/2. - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
E.g.f.: x*(-22 - 4*x + (22 - 18*x + 9*x^2)*exp(x))/2. - G. C. Greubel, Nov 23 2018
Extensions
More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001