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 A155521 Smallest fixed point summed over all non-derangement permutations of {1,2,...,n}. 9
 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:0904.2792 [math.CO], 2009. FORMULA a(n) = (n+1)*a(n-1) +n*(-1)^(n+1); a(0)=0. E.g.f.: (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!*n*(1-2/e). - Vaclav Kotesovec, Oct 20 2012 a(n) = Sum_{k=0..n-1} (k+1) * A047920(n-1,k). - Alois P. Heinz, Sep 01 2021 D-finite with recurrence a(n) +(-n+1)*a(n-1) +(-2*n+1)*a(n-2) +(-n+1)*a(n-3)=0. - R. J. Mathar, Jul 26 2022 EXAMPLE a(3)=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 A329944 A060015 Adjacent sequences: A155518 A155519 A155520 * A155522 A155523 A155524 KEYWORD nonn AUTHOR Emeric Deutsch, Apr 21 2009 STATUS approved

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Last modified November 30 04:37 EST 2022. Contains 358431 sequences. (Running on oeis4.)