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A000565 Number of discordant permutations.

%I A000565 M5227 N2275 #38 Sep 08 2022 08:44:28
%S A000565 31,696,5823,29380,108933,327840,848380,1958004,4130895,8107024,
%T A000565 14990889,26372124,44470165,72305160,113897310,174496828,260846703,
%U A000565 381480456,547057075,770735316,1068589557,1460069392,1968505152,2621661540
%N A000565 Number of discordant permutations.
%D A000565 J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
%D A000565 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000565 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000565 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000565 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000565 J. Riordan, <a href="/A000211/a000211.pdf">Discordant permutations</a>, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
%H A000565 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28, 56,-70,56,-28,8,-1).
%F A000565 From Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001: (Start)
%F A000565 G.f.: -x^7(12x^7 - 6x^6 + 88x^5 - 131x^4 - 548x^3 - 1123x^2 - 448x - 31) / ((1 - x)^8).
%F A000565 a(n) = 243/560n^7 - 243/16n^6 + 3591/16n^5 - 28737/16n^4 + 82257/10n^3 - 81931/4n^2 + 151931/7n, for n>6. (End)
%p A000565 pp := n - >243/560*n^7 - 243/16*n^6 + 3591/16*n^5 - 28737/16*n^4 + 82257/10*n^3 - 81931/4*n^2 + 151931/7*n; seq(pp(n), n=0..30); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
%p A000565 A000565:=-(12*z**7-6*z**6-131*z**4+88*z**5-1123*z**2-548*z**3-31-448*z)/(z-1)**8; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A000565 LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {31, 696, 5823, 29380, 108933, 327840, 848380, 1958004}, 30] (* _Jean-François Alcover_, Feb 10 2016 *)
%o A000565 (Magma) [243/560*n^7-243/16*n^6+3591/16*n^5-28737/16*n^4+ 82257/10*n^3 - 81931/4*n^2+151931/7*n: n in [7..45]]; // _Vincenzo Librandi_, Feb 10 2016
%K A000565 nonn,easy
%O A000565 7,1
%A A000565 _N. J. A. Sloane_
%E A000565 More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001