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A186374 Number of strong fixed blocks in all the permutations of [n] (see first comment for definition).

%I A186374 #32 Aug 05 2015 17:10:38
%S A186374 0,1,1,3,11,48,248,1500,10476,83328,745344,7413120,81187200,970928640,
%T A186374 12589240320,175900757760,2634526944000,42103369728000,
%U A186374 715107004416000,12862666543104000,244249409359872000,4882687056543744000,102496533840691200000
%N A186374 Number of strong fixed blocks in all the permutations of [n] (see first comment for definition).
%C A186374 A fixed block of a permutation p is a maximal sequence of consecutive fixed points of p. For example, the permutation 213486759 has 3 fixed blocks: 34, 67, and 9. A fixed block f of a permutation p is said to be strong if all the entries to the left (right) of f are smaller (larger) than all the entries of f. In the above example, only 34 and 9 are strong fixed blocks.
%H A186374 Alois P. Heinz, <a href="/A186374/b186374.txt">Table of n, a(n) for n = 0..450</a>
%F A186374 a(n) = Sum(k*A186373(n,k), k>=0).
%F A186374 Apparently, a(n) = A003149(n-1)-A003149(n-2) or, equivalently, a(n)=(n-1)! + Sum(k!*(n-2-k)!*(n-2-k), k=0..n-2).
%F A186374 a(n) ~ 2 * (n-1)! * ((1 + 1/n^2 + 7/n^3 + 49/n^4 + 391/n^5 + 3601/n^6 + 37927/n^7 + 451249/n^8 + 5995591/n^9 + 88073041/n^10)). - _Vaclav Kotesovec_, Mar 17 2015
%F A186374 Recurrence (for n>=3): 2*(n^2 - 7*n + 11)*a(n) = (n-2)*(3*n^2 - 17*n + 17)*a(n-1) - (n-2)^2*(n^2 - 5*n + 5)*a(n-2). - _Vaclav Kotesovec_, Aug 04 2015
%e A186374 a(3) = 3 because in [123], [1]32, 21[3], 231, 312, 321 we have 1 + 1 + 1 + 0 + 0 + 0 strong fixed blocks (shown between square brackets).
%p A186374 a:= proc(n) option remember; `if`(n<5, [0, 1, 1, 3, 11][n+1],
%p A186374       ((3*n^2-12*n+2)*a(n-1) -(n^3-3*n^2-8*n+23)*a(n-2)
%p A186374        +(n-3)^3*a(n-3)) / (2*n-8))
%p A186374     end:
%p A186374 seq(a(n), n=0..24);  # _Alois P. Heinz_, May 22 2013
%t A186374 Flatten[{0, 1, Table[(n-1)! + Sum[k!*(n-2-k)!*(n-2-k), {k,0,n-2}], {n,2,20}]}] (* _Vaclav Kotesovec_, Aug 04 2015 *)
%t A186374 Flatten[{0, Simplify[Table[Gamma[n] * (1 - (n-2)*(I*Pi/2^n + LerchPhi[2, 1, n])), {n, 1, 20}]]}] (* _Vaclav Kotesovec_, Aug 04 2015 *)
%Y A186374 Cf. A186373, A003149.
%K A186374 nonn
%O A186374 0,4
%A A186374 _Emeric Deutsch_, Apr 18 2011
%E A186374 a(11)-a(22) from _Alois P. Heinz_, May 22 2013