|
%I #28 Jul 12 2022 08:40:11
%S 1,1,4,12,57,321,2176,17008,150505,1485465,16170036,192384876,
%T 2483177809,34554278857,515620794592,8212685046336,139062777326001,
%U 2494364438359953,47245095998005060,942259727190907180,19737566982241851721,433234326593362631601
%N Number of permutations of {1,2,...,n} having exactly one string of consecutive fixed points (including singletons).
%C Empirically the partial sums of A000240. - _Sean A. Irvine_, Jul 12 2022
%H Alois P. Heinz, <a href="/A177265/b177265.txt">Table of n, a(n) for n = 1..450</a>
%F a(n) = (1/2)[1-(-1)^n] + Sum_{j=1..n} d[j], where d[j] = A000166(n) are the derangement numbers.
%F a(1) = 1, a(2) = 1, a(n) = a(n-1) + n*A000166(n-1). - _Daniel Suteu_, Jan 25 2018
%F Conjecture: D-finite with recurrence a(n) +(-n+1)*a(n-1) +(-n+1)*a(n-2) +(n-1)*a(n-3) +(n-2)*a(n-4)=0. - _R. J. Mathar_, Jul 01 2022
%e a(4,1) = 12 because we have (the string of consecutive fixed points is between square brackets): [1]342, [1]423, [12]43, [1234], 3[2]41, 4[2]13, 4[23]1, 24[3]1, 41[3]2, 21[34], 231[4], and 312[4].
%p d := proc (n) options operator, arrow: factorial(n)*(sum((-1)^i/factorial(i), i = 0 .. n)) end proc: a := proc (n) options operator, arrow: 1/2-(1/2)*(-1)^n+add(d(j), j = 1 .. n) end proc; seq(a(n), n = 1 .. 22);
%t a[0] = 1; a[n_] := a[n] = n*a[n - 1] + (-1)^n; f[n_] := Sum[(n - k) a[n - k - 1], {k, 0, n - 1}]; Array[f, 20] (* _Robert G. Wilson v_, Apr 01 2011 *)
%Y Cf. A000166. Column A180192(n,1).
%Y Cf. A000240.
%K nonn
%O 1,3
%A _Emeric Deutsch_, May 25 2010
|
