[seqfan] Permutations with 2K-2 odd displacements, formula?

[Ron Hardin]

Can the formula for this be guessed?  The (full) row sums are n! of course.

/tmp/dhp
T(n,k)=Number of permutations of n elements with 2k-2 odd displacements

...........1.............0.............0............0...........0.........0......0

...........1.............1.............0............0...........0.........0......0

...........2.............4.............0............0...........0.........0......0

...........4............16.............4............0...........0.........0......0

..........12............72............36............0...........0.........0......0

..........36...........324...........324...........36...........0.........0......0

.........144..........1728..........2592..........576...........0.........0......0

.........576..........9216.........20736.........9216.........576.........0......0

........2880.........57600........172800.......115200.......14400.........0......0

.......14400........360000.......1440000......1440000......360000.....14400......0

.......86400.......2592000......12960000.....17280000.....6480000....518400......0

......518400......18662400.....116640000....207360000...116640000..18662400.518400

.....3628800.....152409600....1143072000...2540160000..1905120000.457228800.......

....25401600....1244678400...11202105600..31116960000.31116960000.................

...203212800...11379916800..119489126400.398297088000.............................

..1625702400..104044953600.1274550681600..........................................

.14631321600.1053455155200........................................................


column 1 is A010551

All solutions for n=4 k=2
..3....1....1....3....0....1....0....0....3....2....2....2....0....1....3....2
..1....3....3....1....2....2....1....3....2....0....0....1....2....0....0....3
..0....0....2....2....1....0....3....1....0....1....3....3....3....2....2....1
..2....2....0....0....3....3....2....2....1....3....1....0....1....3....1....0


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