%I A000352 M3954 N1629
%S A000352 5,29,118,418,1383,4407,13736,42236,128761,390385,1179354,3554454,
%T A000352 10696139,32153963,96592972,290041072,870647517,2612991141,7841070590,
%U A000352 23527406090,70590606895,211788597919,635399348208,1906265153508
%N A000352 One half of the number of permutations of [n] such that the differences 
               have three runs with the same signs.
%D A000352 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000352 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000352 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A000352 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #13
%D A000352 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 260.
%H A000352 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000352 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000352 E. Rodney Canfield and Herbert S. Wilf, <a href="http://arXiv.org/abs/
               math.CO/0609704">Counting permutations by their runs up and down</
               a>
%F A000352 G.f.: (5-6*x)/((1-3*x)*(1-2*x)*(1-x)^2).
%F A000352 4*a(n)/3^n ->1 as n ->infinity . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 22 2004
%e A000352 a(4)=5 because the permutations of [4] with three sign runs are 1324, 
               1423, 2143, 2314, 2413 and their reversals.
%p A000352 A000352:=-(-5+6*z)/(3*z-1)/(2*z-1)/(z-1)**2; [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%p A000352 a:= n-> (Matrix([[0,0,1,2]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif 
               j=1 then [7,-17,17,-6][i] else 0 fi)^n)[1,4]; seq (a(n), n=4..27); 
               [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 26 2008]
%Y A000352 a(n)=T(n, 3), where T(n, k) is the array defined in A008970. Cf. A000486, 
               A000506.
%Y A000352 Sequence in context: A085151 A119494 A153077 this_sequence A034332 A146053 
               A163082
%Y A000352 Adjacent sequences: A000349 A000350 A000351 this_sequence A000353 A000354 
               A000355
%K A000352 nonn
%O A000352 4,1
%A A000352 N. J. A. Sloane (njas(AT)research.att.com).
%E A000352 Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2004