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%I #53 Jul 15 2022 13:36:49
%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.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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