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A177265 Number of permutations of {1,2,...,n} having exactly one string of consecutive fixed points (including singletons). 3
1, 1, 4, 12, 57, 321, 2176, 17008, 150505, 1485465, 16170036, 192384876, 2483177809, 34554278857, 515620794592, 8212685046336, 139062777326001, 2494364438359953, 47245095998005060, 942259727190907180, 19737566982241851721, 433234326593362631601 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Empirically the partial sums of A000240. - Sean A. Irvine, Jul 12 2022
LINKS
FORMULA
a(n) = (1/2)[1-(-1)^n] + Sum_{j=1..n} d[j], where d[j] = A000166(n) are the derangement numbers.
a(1) = 1, a(2) = 1, a(n) = a(n-1) + n*A000166(n-1). - Daniel Suteu, Jan 25 2018
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
EXAMPLE
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].
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
Cf. A000166. Column A180192(n,1).
Cf. A000240.
Sequence in context: A065125 A208940 A209068 * A243923 A192331 A068525
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 25 2010
STATUS
approved

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Last modified May 20 19:00 EDT 2024. Contains 372720 sequences. (Running on oeis4.)