%I A097971
%S A097971 2,10,56,360,2640,21840,201600,2056320,22982400,279417600,3672345600,
%T A097971 51891840000,784604620800,12640852224000,216202162176000,
%U A097971 3912561709056000,74694359900160000,1500289571708928000
%N A097971 Number of alternating runs in all permutations of [n] (the permutation 
               732569148 has four alternating runs: 732, 2569, 91 and 148).
%D A097971 M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, 
               FL, 2004, pp. 24-30.
%D A097971 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, 
               MA, 1973, Vol. 3, pp. 46 and 587-8.
%F A097971 a(n)=n!(2n-1)/3. E.g.f. = x^2*(3-x)/[3(1-x)^2]. a(n)=2*A006157
%e A097971 a(3)=10 because the permutations 123, 132, 312, 213, 231, 321 have the 
               following alternating runs: 123, 13, 32, 31, 12, 21, 13, 23, 31 and 
               321.
%p A097971 seq(n!*(2*n-1)/3,n=2..20);
%Y A097971 Cf. A006157.
%Y A097971 Sequence in context: A108490 A165817 A000172 this_sequence A093303 A075870 
               A074608
%Y A097971 Adjacent sequences: A097968 A097969 A097970 this_sequence A097972 A097973 
               A097974
%K A097971 nonn
%O A097971 2,1
%A A097971 Emeric Deutsch and Ira Gessel (deutsch(AT)duke.poly.edu), Sep 07 2004