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 A000475 Rencontres numbers: number of permutations of [n] with exactly 4 fixed points. (Formerly M4969 N2132) 21
 1, 0, 15, 70, 630, 5544, 55650, 611820, 7342335, 95449640, 1336295961, 20044438050, 320711010620, 5452087178160, 98137569209940, 1864613814984984, 37292276299704525, 783137802293789040, 17229031650463366195, 396267727960657413630 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,3 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65. 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 T. D. Noe, Table of n, a(n) for n=4..100 FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation FORMULA a(n) = sum((-1)^j*n!/(4!*j!), j=2..n-4). a(n) = A000166(n)*binomial(n+4, 4). - Robert Goodhand (robert(AT)rgoodhand.fsnet.co.uk), Nov 08 2001 E.g.f.: (exp(-x)/(1-x))*(x^4/4!). In general, for k fixed points:(exp(-x)/(1-x)) * (x^k/k!). - Wenjin Woan, Nov 22 2008 a(n) ~ n! * exp(-1)/24, in general a(n) ~ n! * exp(-1)/k!. - Vaclav Kotesovec, Mar 16 2014 a(n) = n*a(n-1) + (-1^n)*binomial(n,4) with a(n) = 0 for n = 0,1,2,3. - Chai Wah Wu, Nov 01 2014 Conjecture: (-n+4)*a(n) +n*(n-5)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 02 2015 O.g.f.: (1/24)*Sum_{k>=4} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017 MAPLE a:=n->sum(n!*sum((-1)^k/(k-3)!, j=0..n), k=3..n): seq(-a(n)/4!, n=3..22); # Zerinvary Lajos, May 25 2007 G(x):=exp(-x)/(1-x)*(x^4/4!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=4..23); # Zerinvary Lajos, Apr 03 2009 MATHEMATICA Table[Subfactorial[n - 4]*Binomial[n, 4], {n, 4, 23}] (* Zerinvary Lajos, Jul 10 2009 *) PROG (PARI) x='x+O('x^66); Vec( serlaplace(exp(-x)/(1-x)*(x^4/4!)) ) \\ Joerg Arndt, Feb 19 2014 (Python) from sympy import binomial A000475_list, m, x = [], 1, 0 for n in range(4, 100): ....x, m = x*n + m*binomial(n, 4), -m ....A000475_list.append(x) # Chai Wah Wu, Nov 01 2014 CROSSREFS Cf. A008290, A000166, A000240, A000387, A000449, A129135. A diagonal of A008291. Cf. A170942. Sequence in context: A124893 A126402 A053134 * A253476 A145053 A168298 Adjacent sequences:  A000472 A000473 A000474 * A000476 A000477 A000478 KEYWORD nonn AUTHOR EXTENSIONS Formula corrected by Sean A. Irvine, Oct 26 2010 STATUS approved

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