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%I #11 Jul 04 2019 03:41:43
%S 0,0,1,4,22,138,998,8174,74898,759634,8451862,102381222,1341503546,
%T 18907621562,285259758366,4587192222958,78327809126818,
%U 1415429225667234,26987142531214118,541434621007942454,11402270678456333322
%N Number of increasing runs of even length in all permutations of [n].
%H Vincenzo Librandi, <a href="/A097593/b097593.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: (4*(exp(-x)-1)+4*x-x^2)/(2*(1-x)^2).
%F 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
%F a(n) ~ n!*n*(4*exp(-1)-1)/2. - _Vaclav Kotesovec_, Nov 19 2012
%F a(n) = Sum_{k=1..floor(n/2)} k * A097592(n,k). - _Alois P. Heinz_, Jul 04 2019
%e Example: a(3)=4 because we have 123,(13)2,2(13),(23)1,3(12),321 (runs of even length shown between parentheses).
%p 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);
%t 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 *)
%o (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
%Y Cf. A097592.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Aug 29 2004
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