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 A000266 E.g.f. exp(-x^2/2) / (1-x). (Formerly M2991 N1211) 16
 1, 1, 1, 3, 15, 75, 435, 3045, 24465, 220185, 2200905, 24209955, 290529855, 3776888115, 52876298475, 793144477125, 12690313661025, 215735332237425, 3883235945814225, 73781482970470275, 1475629660064134575, 30988222861346826075, 681740902935880863075 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no transposition. REFERENCES Carter, Larry, and Stan Wagon. "The Mensa Correctional Institute." The American Mathematical Monthly 125.4 (2018): 306-319. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85. 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). R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe) INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 104 Simon Plouffe, Exact formulas for integer sequences, March 1993. FORMULA E.g.f.: exp( x + Sum_{k>2} x^k / k ). - Michael Somos, Jul 25 2011 a(n) = n! * sum i=0 ... [n/2]( (-1)^i /(i! * 2^i)); a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 2^i) = e^(-1/2); a(n) ~ e^(-1/2) * n!; a(n) ~ e^(-1/2) * (n/e)^n * sqrt(2 * Pi * n). - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001 A027616(n) + a(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003 a(n) = n!*floor((floor(n/2)! * 2^floor(n/2) / exp(1/2)+1/2)) / floor(n/2)! / 2^floor(n/2), n>=0. - Simon Plouffe from old notes, 1993 E.g.f.: 1/(1-x)*exp(-(x^2)/2) = 1/((1-x)*G(0)); G(k)= 1+(x^2)/(2*(2*k+1)-2*(x^2)*(2*k+1)/((x^2)+4*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011 E.g.f.: 1/Q(0), where Q(k)= 1 - x/(1 - x/(x - (2*k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013 EXAMPLE a(3) = 3 because the permutations in S_3 that contain no transpositions are the trivial permutation and the two 3-cycles. MAPLE G:=exp(-z^2/2)/(1-z): Gser:=series(G, z=0, 26): for n from 0 to 25 do a(n):=n!*coeff(Gser, z, n): end do: seq(a(n), n=0..20); # Paul Weisenhorn, May 29 2010 # second Maple program: a:= proc(n) option remember; `if`(n=0, 1, add(       a(n-j)*(j-1)!*binomial(n-1, j-1), j=[1, \$3..n]))     end: seq(a(n), n=0..30);  # Alois P. Heinz, May 12 2016 MATHEMATICA a=Log[1/(1-x)]-x^2/2; Range[0, 20]! CoefficientList[Series[Exp[a], {x, 0, 20}], x] (* Geoffrey Critzer, Nov 29 2011 *) PROG (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp(-(x^2/2)+x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */ CROSSREFS See also A000138 and A000090. Cf. A027616, A130905, A193385. Sequence in context: A136778 A300665 A278398 * A294340 A059838 A079164 Adjacent sequences:  A000263 A000264 A000265 * A000267 A000268 A000269 KEYWORD nonn AUTHOR EXTENSIONS More terms from Christian G. Bower Entry improved by comments from Michael Somos, Jul 28 2009 Minor editing by Johannes W. Meijer, Jul 25 2011 STATUS approved

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Last modified July 18 23:45 EDT 2018. Contains 312766 sequences. (Running on oeis4.)