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 A001189 Number of degree-n permutations of order exactly 2. (Formerly M2801 N1127) 40
 0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503, 2390479, 10349535, 46206735, 211799311, 997313823, 4809701439, 23758664095, 119952692895, 618884638911, 3257843882623, 17492190577599, 95680443760575, 532985208200575, 3020676745975551 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Number of set partitions of [n] into blocks of size 2 and 1 with at least one block of size 2. - Olivier Gérard, Oct 29 2007 For n>=2, number of standard Young tableaux with height <= n - 1. That is, all tableaux (A000085) but the one with just one column. - Joerg Arndt, Oct 24 2012 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..800 N. Chheda, M. K. Gupta, RNA as a Permutation, arXiv:1403.5477 [q-bio.BM], 2014. R. B. Herrera, The number of elements of given period in finite symmetric group, Amer. Math. Monthly 64, 1957, 488-490. L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168. J. Rangel-Mondragon, Selected Themes in Computational Non-Euclidean Geometry: Part 1, The Mathematica Journal 15 (2013); http://www.mathematica-journal.com/data/uploads/2013/07/Rangel-Mondragon_Selected-1.pdf Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., April 2004. FORMULA E.g.f.: exp(x + x^2/2) - exp(x). a(n) = A000085(n) - 1. a(n) = b(n, 2), where b(n, d)=Sum_{k=1..n} (n-1)!/(n-k)! * Sum_{l:lcm{k, l}=d} b(n-k, l), b(0, 1)=1 is the number of degree-n permutations of order exactly d. From Henry Bottomley, May 03 2001: (Start) a(n) = a(n-1) + (1 + a(n-2))*(n-1). a(n) = Sum_{j=1..floor(n/2)} n!/(j!*(n-2*j)!*(2^j)). (End) MAPLE a:= proc(n) option remember; `if`(n<3, [0\$2, 1][n+1],       a(n-1) +(n-1) *(1+a(n-2)))     end: seq(a(n), n=1..30);  # Alois P. Heinz, Oct 24 2012 # alternative: A001189 := proc(n)     local a, prs, p, k ;     a := 0 ;     for prs from 1 to n/2 do         p := product(binomial(n-2*k, 2), k=0..prs-1) ;         a := a+p/prs!;     end do:     a; end proc: seq(A001189(n), n=1..13) ; # R. J. Mathar, Jan 04 2017 MATHEMATICA RecurrenceTable[{a[1]==0, a[2]==1, a[n]==a[n-1]+(1+a[n-2])(n-1)}, a[n], {n, 25}] (* Harvey P. Dale, Jul 27 2011 *) PROG (PARI) {a(n) = sum(j=1, floor(n/2), n!/(j!*(n-2*j)!*2^j))}; \\ G. C. Greubel, May 14 2019 (MAGMA) m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2) -Exp(x) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, May 14 2019 (Sage) m = 30; T = taylor(exp(x +x^2/2) - exp(x), x, 0, m); a=[factorial(n)*T.coefficient(x, n) for n in (0..m)]; a[1:] # G. C. Greubel, May 14 2019 CROSSREFS Cf. A001470-A001473, A052501, A053496-A053504, A061121-A061128. Column k=1 of A143911, column k=2 of A080510, A182222. - Alois P. Heinz, Oct 24 2012 Column k=2 of A057731. - Alois P. Heinz, Feb 14 2013 Sequence in context: A183111 A132835 A191354 * A212352 A198180 A101786 Adjacent sequences:  A001186 A001187 A001188 * A001190 A001191 A001192 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 16 11:23 EDT 2019. Contains 328056 sequences. (Running on oeis4.)