4,1

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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.

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

E. Rodney Canfield and Herbert S. Wilf, Counting permutations by their runs up and down

G.f.: (5-6*x)/((1-3*x)*(1-2*x)*(1-x)^2).

4*a(n)/3^n ->1 as n ->infinity . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 22 2004

a(4)=5 because the permutations of [4] with three sign runs are 1324, 1423, 2143, 2314, 2413 and their reversals.

A000352:=-(-5+6*z)/(3*z-1)/(2*z-1)/(z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]

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]

a(n)=T(n, 3), where T(n, k) is the array defined in A008970. Cf. A000486, A000506.

Adjacent sequences: A000349 A000350 A000351 this_sequence A000353 A000354 A000355

Sequence in context: A085151 A119494 A153077 this_sequence A034332 A146053 A163082

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2004