A308962 Number of permutations of [4n] with exactly 2n increasing runs of odd length.

1, 17, 13930, 77296296, 1568558071080, 84938094880524600, 10128482222614148352960, 2336936362896740255803152000, 950622895076910219544822877635200, 635598214592375283010356491822548022400, 661314598267382330509313757278639302452192000

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..110 (terms 0..69 from Alois P. Heinz)

Formula

a(n) = (4n)! * [x^(4n) t^(2n)] t^2/(1-t*x-(1-t^2)*exp(-t*x)). [corrected by Vaclav Kotesovec, Jul 09 2019]

a(n) = A097591(4n,2n).

From Vaclav Kotesovec, Jul 09 2019: (Start)

a(n)/(4*n)! ~ c * d^n / sqrt(n), where

d = 0.49313160144517183347479521733129940030484540928084707469774969650583707...

c = 3.44699229707824751737600849250650265725079793249740793784564520854062204...

a(n) ~ c * d^n * n^(4*n), where

d = 2.31219720619339615667811172118287009649702081583503593066663730992576726...

c = 17.2806567085831933774093124549232969200598807738253988225436890867215712...

(End)

Example

a(1) = 17: (124)(3), (134)(2), 14(3)(2), (2)(134), (2)14(3), (234)(1), 24(3)(1), (3)(124), (3)14(2), (3)(2)14, (3)24(1), 34(2)(1), (4)(123), (4)13(2), (4)(2)13, (4)23(1), (4)(3)12; (odd length runs are shown between parentheses).

Mathematica

Flatten[{1, Table[(4 n)! * Coefficient[Expand[Normal[Series[t^2/(1 - t*x - (1 - t^2)*E^(-t*x)), {x, 0, 4*n}, {t, 0, 2*n}]]], x^(4*n)*t^(2*n)], {n, 1, 10}]}] (* Vaclav Kotesovec, Jul 09 2019 *)

Crossrefs

Cf. A097591.

Sequence in context: A114924 A362066 A300596 * A191964 A230638 A308671

Adjacent sequences: A308959 A308960 A308961 * A308963 A308964 A308965

Keyword

nonn

Author

Alois P. Heinz, Jul 03 2019

Status

approved