A152666 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of odd entries (1<=k<=ceiling(n/2)). For example, the permutation 321756498 has 3 runs of odd entries: 3, 175 and 9.

1, 2, 4, 2, 12, 12, 36, 72, 12, 144, 432, 144, 576, 2592, 1728, 144, 2880, 17280, 17280, 2880, 14400, 115200, 172800, 57600, 2880, 86400, 864000, 1728000, 864000, 86400, 518400, 6480000, 17280000, 12960000, 2592000, 86400, 3628800, 54432000

Offset

1,2

Comments

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

Row n contains ceiling(n/2) entries.

T(n,1) = A010551(n+1).

Sum_{k>=1} k*T(n,k) = A052618(n-1).

Mirror image of A134435.

Links

Table of n, a(n) for n=1..38.

Formula

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

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

Example

T(3,2)=2 because we have 123 and 321.
T(4,2)=12 because we have 1234, 1432, 3214, 3412, 1243, 3241 and their reverses.
Triangle starts:
1;
2;
4,2;
12,12;
36,72,12;
144,432,144;
576,2592,1728,144.

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, k-1)*binomial(n+1, 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 .. ceil((1/2)*n)) end do; # yields sequence in triangular form

Mathematica

T[n_?EvenQ, k_] := (n/2)!^2*Binomial[n/2 - 1, k - 1]*Binomial[n/2 + 1, k]; T[n_?OddQ, k_] := ((n - 1)/2 + 1)!*((n - 1)/2)!*Binomial[(n - 1)/2 + 1, k]*Binomial[(n - 1)/2, k - 1]; Table[T[n, k], {n, 1, 12}, {k, 1, Floor[(n + 1)/2]}] // Flatten (* Jean-François Alcover, Nov 13 2016 *)

Crossrefs

Cf. A000142, A010551, A052618, A152667, A134435.

Sequence in context: A161795 A138770 A006018 * A153801 A354408 A062867

Adjacent sequences: A152663 A152664 A152665 * A152667 A152668 A152669

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 14 2008

Status

approved