A152877 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k consecutive triples of the form (odd,even,odd) and (even,odd,even) (0<=k<=n-2).

1, 1, 2, 4, 2, 16, 0, 8, 60, 24, 24, 12, 288, 144, 216, 0, 72, 1584, 1296, 1152, 576, 288, 144, 10368, 9216, 10368, 4608, 4608, 0, 1152, 74880, 83520, 86400, 60480, 31680, 17280, 5760, 2880, 604800, 748800, 892800, 576000, 460800, 172800, 144000, 0, 28800

Offset: 0

Author

Emeric Deutsch, Dec 17 2008

Keywords

• nonn
• tabf

Comments

Row n has n-1 entries (n>=2).

Sum of entries in row n is n! (A000142(n)).

T(n,0) = A152876(n).

T(n,n-2) = A092186(n).

T(2n+1,2n-2) = A047677(n) = 2*n!*(n+1)!. - Alois P. Heinz, Nov 10 2013

Examples

			T(3,1) = 2 because we have 123 and 321.
Triangle starts:
      1;
      1;
      2;
      4,    2;
     16,    0,     8;
     60,   24,    24,   12;
    288,  144,   216,    0,   72;
   1584, 1296,  1152,  576,  288, 144;
  10368, 9216, 10368, 4608, 4608,   0, 1152;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..142, flattened
• E. Munarini and N. Zagaglia Salvi, Binary strings without zigzags, Sem. Lotharingien de Combinatoire, 49, 2004, B49h.

Crossrefs

Cf. A000142, A047677, A152876, A092186, A329550.

Programs

Maple

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, expand(
      o*b(o-1, u, [2, 2, 5, 5, 2][t])*`if`(t=4, x, 1)+
      u*b(o, u-1, [3, 4, 3, 3, 4][t])*`if`(t=5, x, 1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
               b(ceil(n/2), floor(n/2), 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 10 2013

Mathematica

b[o_, u_, t_] := b[o, u, t] = If[u+o == 0, 1, Expand[o*b[o-1, u, {2, 2, 5, 5, 2}[[t]]]*If[t == 4, x, 1] + u*b[o, u-1, {3, 4, 3, 3, 4}[[t]]]*If[t == 5, x, 1]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] [b[Ceiling[n/2], Floor[n/2], 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

Formula

It would be good to have a formula or generating function for this sequence (a formula for column 0 is given in A152876).

Sum_{k>=1} k*T(n,k) = A329550(n). - Alois P. Heinz, Nov 16 2019

Extensions

More terms from Alois P. Heinz, Nov 10 2013