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

Author

Emeric Deutsch, Aug 29 2004

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A097592.

Programs

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 *)

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

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