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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).
5
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
0,3
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
LINKS
E. Munarini and N. Zagaglia Salvi, Binary strings without zigzags, Sem. Lotharingien de Combinatoire, 49, 2004, B49h.
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
EXAMPLE
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;
...
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 17 2008
EXTENSIONS
More terms from Alois P. Heinz, Nov 10 2013
STATUS
approved