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A155521 Smallest fixed point summed over all non-derangement permutations of {1,2,...,n}. 4
0, 1, 1, 7, 31, 191, 1331, 10655, 95887, 958879, 10547659, 126571919, 1645434935, 23036089103, 345541336531, 5528661384511, 93987243536671, 1691770383660095, 32143637289541787, 642872745790835759, 13500327661607550919 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is also the number of permutations of {1,2,...,n,n+1} having at least 2 fixed points. Example: a(3)=7 because we have 1234, 1243, 1324, 1432, 2134, 4231, and 3214.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792v1, 2009.

FORMULA

Rec. rel: a(n) = (n+1)*a(n-1) +n*(-1)^(n+1); a(0)=0.

Egf = [1-(1+x^2)*exp(-x)]/(1-x)^2.

a(n)=(n+1)!+(-1)^n-2(n+1)d(n),

a(n)=(n+1)!-(n+1)d(n)-d(n+1), where d(n)=A000166(n) are the derangement numbers. a(n)=(n+1)!+(-1)^n -2(n+1)d(n)

a(n) ~ n!*n*(1-2/e). - Vaclav Kotesovec, Oct 20 2012

EXAMPLE

a(n)=7 because the non-derangements of {1,2,3} are 123, 132, 213, 321 with smallest fixed points 1, 1, 3, 2.

MAPLE

a[0] := 0: for n to 25 do a[n] := (n+1)*a[n-1]+n*(-1)^(n+1) end do: seq(a[n], n = 0 .. 21);

MATHEMATICA

CoefficientList[Series[(1-(1+x^2)*E^(-x))/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

CROSSREFS

Cf. A000166, A047920

Sequence in context: A139060 A324621 A223144 * A201116 A060015 A261558

Adjacent sequences:  A155518 A155519 A155520 * A155522 A155523 A155524

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 21 2009

STATUS

approved

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Last modified October 20 22:37 EDT 2019. Contains 328291 sequences. (Running on oeis4.)