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 A161130 Sum of the differences between the largest and the smallest fixed points over all non-derangement permutations of {1,2,...,n}. 2
 0, 0, 1, 2, 13, 74, 523, 4178, 37609, 376082, 4136911, 49642922, 645357997, 9035011946, 135525179203, 2168402867234, 36862848742993, 663531277373858, 12607094270103319, 252141885402066362, 5294979593443393621 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792v1 [math.CO], 2009. FORMULA E.g.f.: (exp(-x) * (1+x+x^2) - 1) / (1-x)^2. a(n) = A000166(n+1) - A155521(n). a(n) = Sum(k*A161129(n,k), k=0..n-1). Recurrence: (n-2)*a(n) = (n^2-2*n-1)*a(n-1) + (n-1)*n*a(n-2). - Vaclav Kotesovec, Oct 20 2012 a(n) ~ n!*n*(3/e-1). - Vaclav Kotesovec, Oct 20 2012 EXAMPLE a(3)=2 because the non-derangements of {1,2,3} are 1'23', 1'32, 213', and 32'1 with differences between the largest and smallest fixed points (marked) equal to 2, 0, 0, and 0, respectively. a(4)=13 because the non-derangements of {1,2,3,4} are 1'234', 1'2'43, 1'423, 1'324', 1'342, 1'43'2, 413'2, 3124', 213'4', 42'13, 2314', 243'1, 42'3'1, 32'14', and 32'41 with differences between the largest and smallest fixed points (marked) equal to 3, 1, 0, 3, 0, 2, 0, 0, 1, 0, 0, 0, 1, 2, and 0, respectively. MAPLE G := (exp(-x)*(1+x+x^2)-1)/(1-x)^2: Gser := series(G, x = 0, 25): seq(factorial(n)*coeff(Gser, x, n), n = 0 .. 22); MATHEMATICA CoefficientList[Series[(E^(-x)*(1+x+x^2)-1)/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *) CROSSREFS Cf. A000166, A000240, A155521, A161129 Sequence in context: A263840 A307288 A154357 * A192700 A007509 A077413 Adjacent sequences:  A161127 A161128 A161129 * A161131 A161132 A161133 KEYWORD nonn AUTHOR Emeric Deutsch, Jul 18 2009 STATUS approved

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Last modified March 29 04:13 EDT 2020. Contains 333105 sequences. (Running on oeis4.)