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A180195 a(n) = (-1)^n*Sum((-1)^j*b(j), j=1..n), where b(n)=(n-1)!*(n^2 - n + 1) = A001564(n-1) (n>=1).

%I A180195 #10 May 19 2024 08:11:14
%S A180195 1,2,12,66,438,3282,27678,259602,2683758,30338322,372458478,
%T A180195 4936475922,70266775278,1069278031122,17325341412078,297824181275922,
%U A180195 5414097458148078,103781942967323922,2092232238097380078,44254551017667611922,979997194424697828078,22675109031076772891922
%N A180195 a(n) = (-1)^n*Sum((-1)^j*b(j), j=1..n), where b(n)=(n-1)!*(n^2 - n + 1) = A001564(n-1) (n>=1).
%C A180195 a(n) is the number of blocks of odd size in all permutations of [n].
%C A180195 a(n) is the number of blocks of even size in all permutations of [n+1].
%C A180195 A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. Example: a(2)=2 because in 12 and (2)(1) we have a total of 2 blocks of odd size (shown between parentheses). Also, in 123, 132, 213, (23)1, 3(12), and 321 we have a total of 2 blocks of even size (shown between parentheses).
%H A180195 A. N. Myers, <a href="https://doi.org/10.1006/jcta.2002.3279">Counting permutations by their rigid patterns</a>, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.
%F A180195 a(n) = Sum(k*A180193(n,k), k>=0).
%F A180195 a(n) = Sum(k*A180194(n+1,k), k>=0).
%F A180195 Conjecture D-finite with recurrence a(n) +(-n-1)*a(n-1) -4*a(n-2) +(n-2)*a(n-3)=0. - _R. J. Mathar_, Jul 24 2022
%p A180195 b := proc (n) options operator, arrow: factorial(n-1)*(n^2-n+1) end proc: a := proc (n) options operator, arrow: (-1)^n*(sum((-1)^j*b(j), j = 1 .. n)) end proc; seq(a(n), n = 1 .. 20);
%o A180195 (PARI) a(n) = (-1)^n*sum(j=1, n, (-1)^j*(j-1)!*(j^2-j+1)); \\ _Michel Marcus_, May 19 2024
%Y A180195 Cf. A001564, A180193, A180194.
%K A180195 nonn
%O A180195 1,2
%A A180195 _Emeric Deutsch_, Sep 09 2010
%E A180195 More terms from _Michel Marcus_, May 19 2024