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Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).
13

%I #56 Apr 27 2024 16:42:33

%S 1,1,14,426,24024,2170680,287250480,52370755920,12585067447680,

%T 3854801333416320,1465957162768492800,677696237345719468800,

%U 374281829360322587827200,243388909697235614324812800,184070135024053703140543027200,160192129141963141211280644352000

%N Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).

%C Also turns up as the solution to Problem #18, p. 326 of Alan Tucker's Applied Combinatorics, 4th ed, Wiley NY 2002 [Tucker's `n' is the `2n' here]. - John L Leonard, Sep 15 2003

%C Number of acyclic orientations of the Turán graph T(2n,n). - _Alois P. Heinz_, Jan 13 2016

%C n-th term of the n-th forward differences of n!. - _Alois P. Heinz_, Feb 22 2019

%D R. P. Stanley, Enumerative Combinatorics I, Chap.2, Exercise 10, p. 89.

%H Reinhard Zumkeller, <a href="/A033815/b033815.txt">Table of n, a(n) for n = 0..200</a>

%H Leo Chao, Paul DesJarlais and John L Leonard, <a href="http://www.jstor.org/stable/3621238">A binomial identity, via derangements</a>, Math. Gaz. 89 (2005), 268-270.

%H Ira Gessel, <a href="http://www.mat.univie.ac.at/~slc/opapers/s17gessel.html">Enumerative applications of symmetric functions</a>, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.

%F a(n) = A002119(n)*n!*(-1)^n.

%F D-finite with recurrence a(n) = 2n*(2n-1)*a(n-1) + n*(n-1)*a(n-2).

%F a(n) = Sum_{i=0..n} binomial(n, i)*(-1)^i*(2*n-i)!.

%F From John L Leonard, Sep 15 2003: (Start)

%F a(n) = Sum_{i=0..n} C(n, i)*(2n-i)!*Sum_{j=0..2n-i} (-1)^j/j!.

%F a(n) = n!*Sum_{i=0..n} C(n, i)*n!/(n-i)!*Sum_{j=0..n-i} (-1)^j*C(n-i, j)*(n-j)!/i!. (End)

%F a(n) = Sum_{k=0..n} binomial(n,k)*A000166(n+k). - _Vladeta Jovovic_, Sep 04 2006

%F a(n) = A116854(2*n+1,n+1). - _Reinhard Zumkeller_, Aug 31 2014

%F a(n) = A267383(2n,n). - _Alois P. Heinz_, Jan 13 2016

%F a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n + 1/2). - _Vaclav Kotesovec_, Feb 18 2017

%F a(n) = n!*exp(-1/2)*((-1)^n * BesselI(n+1/2,1/2)*Pi^(1/2) + BesselK(n+1/2,1/2)/Pi^(1/2) ). - _Mark van Hoeij_, Jul 15 2022

%p A033815 := proc(n) local i; add(binomial(n, i)*(-1)^i*(2*n - i)!, i = 0 .. n) end;

%p # second Maple program:

%p A:= proc(n, k) A(n, k):= `if`(k=0, n!, A(n+1, k-1) -A(n, k-1)) end:

%p a:= n-> A(n$2):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Feb 22 2019

%t a[n_] := (2n)!*Hypergeometric1F1[-n, -2n, -1]; Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Jun 13 2012, after _Vladimir Reshetnikov_ *)

%o (Haskell)

%o a033815 n = a116854 (2 * n + 1) (n + 1)

%o -- _Reinhard Zumkeller_, Aug 31 2014

%Y Main diagonal of array in A068106 and of A047920.

%Y Cf. A000142, A002119, A116854, A267383.

%Y Column k=2 of A372326.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_