A097593 Number of increasing runs of even length in all permutations of [n].

0, 0, 1, 4, 22, 138, 998, 8174, 74898, 759634, 8451862, 102381222, 1341503546, 18907621562, 285259758366, 4587192222958, 78327809126818, 1415429225667234, 26987142531214118, 541434621007942454, 11402270678456333322

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f.: (4*(exp(-x)-1)+4*x-x^2)/(2*(1-x)^2).

a(n) = (2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) - (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Nov 19 2012

a(n) ~ n!*n*(4*exp(-1)-1)/2. - Vaclav Kotesovec, Nov 19 2012

a(n) = Sum_{k=1..floor(n/2)} k * A097592(n,k). - Alois P. Heinz, Jul 04 2019

Example

Example: a(3)=4 because we have 123,(13)2,2(13),(23)1,3(12),321 (runs of even length shown between parentheses).

Maple

G:=(4*(exp(-x)-1)+4*x-x^2)/2/(1-x)^2: Gser:=series(G, x=0, 25): 0, seq(n!*coeff(Gser, x^n), n=1..24);

Mathematica

Table[n!*SeriesCoefficient[(4*(E^(-x)-1)+4*x-x^2)/(2*(1-x)^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)

Prog

(PARI) x='x+O('x^66); concat([0, 0], Vec(serlaplace((4*(exp(-x)-1)+4*x-x^2)/(2*(1-x)^2)))) \\ Joerg Arndt, May 11 2013

Crossrefs

Cf. A097592.

Sequence in context: A091638 A142984 A283055 * A188686 A025756 A366119

Adjacent sequences: A097590 A097591 A097592 * A097594 A097595 A097596

Keyword

nonn

Author

Emeric Deutsch, Aug 29 2004

Status

approved