A152667 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of even entries (n >= 2, 1 <= k <= floor(n/2)). For example, the permutation 321756498 has 3 runs of even entries: 2, 64 and 8.

2, 6, 12, 12, 48, 72, 144, 432, 144, 720, 2880, 1440, 2880, 17280, 17280, 2880, 17280, 129600, 172800, 43200, 86400, 864000, 1728000, 864000, 86400, 604800, 7257600, 18144000, 12096000, 1814400, 3628800, 54432000, 181440000, 181440000

Offset

2,1

Comments

Sum of entries in row n is n! (=A000142(n)). Row n contains floor(n/2) entries.

T(n,1) = A152873(n).

Sum_{k>=1} k*T(n,k) = A152668(n).

Mirror image of A145892.

Links

Table of n, a(n) for n=2..35.

Formula

T(2n,k) = (n!)^2 * binomial(n+1,k) binomial(n-1,k-1);

T(2n+1,k) = n!*(n+1)!*binomial(n-1,k-1)*binomial(n+2,k) (n >= 1).

Example

T(4,2) = 12 because we have 1234, 3214, 1432, 3412, 2134, 2314 and their reverses.
Triangle starts:
    2;
    6;
   12,   12;
   48,   72;
  144,  432,  144;
  720, 2880, 1440;

Maple

ae := proc (n, k) options operator, arrow: factorial(n)^2*binomial(n+1, k)*binomial(n-1, k-1) end proc: ao := proc (n, k) options operator, arrow: factorial(n)*factorial(n+1)*binomial(n-1, k-1)*binomial(n+2, k) end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc; for n to 12 do seq(T(n, k), k = 1 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A000142, A152666, A152668, A145892, A152873.

Sequence in context: A278256 A066791 A062723 * A145892 A216429 A250178

Adjacent sequences: A152664 A152665 A152666 * A152668 A152669 A152670

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 14 2008

Status

approved