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A152876 Number of permutations of {1,2,...,n} having no consecutive triples of the form (odd, even, odd) or (even, odd, even). 2
1, 1, 2, 4, 16, 60, 288, 1584, 10368, 74880, 604800, 5356800, 51840000, 544320000, 6147187200, 74579097600, 962415820800, 13241346048000, 192255565824000, 2957575348224000, 47721518530560000, 811595019755520000, 14407079038894080000, 268390402745794560000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Column 0 of A152877.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..460

E. Munarini and N. Z. Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 1-15.

FORMULA

a(2n) = (n!)^2*Sum_{j=0..floor(n/2)} (binomial(n-j, j))^2*(2*n^2 - 6*j*n + 6*j^2)/(n-j)^2;

a(2n+1) = n!*(n+1)!*Sum_{j=1..floor(n/2)} binomial(n+1-j, j)*binomial(n-j, j)*(2*n^2 + 2*n - 6*j*n - 3*j + 6*j^2)/((n+1-j)*(n-j)).

a(n) ~ 2 * 5^(-1/4) * ((1+sqrt(5))/4)^n * n!. - Vaclav Kotesovec, Sep 03 2014

EXAMPLE

a(3) = 4 because we have 132, 213, 231 and 312.

MAPLE

ae := proc (n) options operator, arrow: 2*(sum((n^2-3*n*j+3*j^2)*binomial(n-j, j)^2/(n-j)^2, j = 0 .. floor((1/2)*n))) end proc: ao := proc (n) options operator, arrow: sum(binomial(n+1-k, k)*binomial(n-k, k)*(2*n^2+2*n-6*k*n-3*k+6*k^2)/((n+1-k)*(n-k)), k = 0 .. floor((1/2)*n)) end proc: a := proc (n) if `mod`(n, 2) = 0 then factorial((1/2)*n)^2*ae((1/2)*n) else factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*ao((1/2)*n-1/2) end if end proc: 1, 1, seq(a(n), n = 2 .. 22);

# second Maple program:

b:= proc(o, u, t) option remember; `if`(u+o=0, 1,

      `if`(t=4, 0, o*b(o-1, u, `if`(t=3, 5, 2)))+

      `if`(t=5, 0, u*b(o, u-1, `if`(t=2, 4, 3))))

    end:

a:= n-> b(ceil(n/2), floor(n/2), 1):

seq(a(n), n=0..30);  # Alois P. Heinz, Nov 11 2013

MATHEMATICA

b[o_, u_, t_] := b[o, u, t] = If[u+o == 0, 1, If[t==4, 0, o*b[o-1, u, If[t==3, 5, 2]]] + If[t==5, 0, u*b[o, u-1, If[t==2, 4, 3]]]]; a[n_] := b[Ceiling[n/2], Floor[n/2], 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A152877.

Sequence in context: A275764 A053349 A228626 * A153963 A153960 A339648

Adjacent sequences:  A152873 A152874 A152875 * A152877 A152878 A152879

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 17 2008

STATUS

approved

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Last modified April 16 03:59 EDT 2021. Contains 343030 sequences. (Running on oeis4.)