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 A000352 One half of the number of permutations of [n] such that the differences have three runs with the same signs. (Formerly M3954 N1629) 4
 5, 29, 118, 418, 1383, 4407, 13736, 42236, 128761, 390385, 1179354, 3554454, 10696139, 32153963, 96592972, 290041072, 870647517, 2612991141, 7841070590, 23527406090, 70590606895, 211788597919, 635399348208, 1906265153508, 5718929678273, 17157057470297 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,1 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #13 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260. 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 = 4..400 E. Rodney Canfield and Herbert S. Wilf, Counting permutations by their runs up and down, arXiv:math/0609704 [math.CO], 2006. 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 Index entries for linear recurrences with constant coefficients, signature (7,-17,17,-6). FORMULA a(n) = (3^n-4*2^n-2*n+11)/4, n>=4. - Tim Monahan, Jul 14 2011 G.f.: x^4*(5-6*x)/((1-3*x)*(1-2*x)*(1-x)^2). Limit_{n->infinity} 4*a(n)/3^n = 1. - Philippe Deléham, Feb 22 2004 EXAMPLE a(4)=5 because the permutations of  with three sign runs are 1324, 1423, 2143, 2314, 2413 and their reversals. MAPLE A000352:=-(-5+6*z)/(3*z-1)/(2*z-1)/(z-1)**2; # [Conjectured by Simon Plouffe in his 1992 dissertation.] [correct up to offset] # second Maple program: a:= n-> (<<0|0|1|2>>. <<7|1|0|0>, <-17|0|1|0>, <17|0|0|1>, <-6|0|0|0>>^n)[1, 4]: seq(a(n), n=4..30); # Alois P. Heinz, Aug 26 2008 MATHEMATICA nn = 40; CoefficientList[Series[x^4*(5 - 6*x)/((1 - 3*x)*(1 - 2*x)*(1 - x)^2), {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *) PROG (PARI) a(n) = (3^n-4*2^n-2*n+11)/4; CROSSREFS a(n) = T(n, 3), where T(n, k) is the array defined in A008970. Cf. A000486, A000506. Sequence in context: A268244 A297632 A153077 * A327133 A267921 A241676 Adjacent sequences: A000349 A000350 A000351 * A000353 A000354 A000355 KEYWORD nonn AUTHOR EXTENSIONS Edited by Emeric Deutsch, Feb 18 2004 STATUS approved

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Last modified March 28 03:48 EDT 2023. Contains 361577 sequences. (Running on oeis4.)