A033815 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).

1, 1, 14, 426, 24024, 2170680, 287250480, 52370755920, 12585067447680, 3854801333416320, 1465957162768492800, 677696237345719468800, 374281829360322587827200, 243388909697235614324812800, 184070135024053703140543027200, 160192129141963141211280644352000

Offset

0,3

Comments

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

Number of acyclic orientations of the Turán graph T(2n,n). - Alois P. Heinz, Jan 13 2016

n-th term of the n-th forward differences of n!. - Alois P. Heinz, Feb 22 2019

References

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..200

Leo Chao, Paul DesJarlais and John L Leonard, A binomial identity, via derangements, Math. Gaz. 89 (2005), 268-270.

Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.

Formula

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

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

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

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

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

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)

a(n) = Sum_{k=0..n} binomial(n,k)*A000166(n+k). - Vladeta Jovovic, Sep 04 2006

a(n) = A116854(2*n+1,n+1). - Reinhard Zumkeller, Aug 31 2014

a(n) = A267383(2n,n). - Alois P. Heinz, Jan 13 2016

a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n + 1/2). - Vaclav Kotesovec, Feb 18 2017

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

Maple

A033815 := proc(n) local i; add(binomial(n, i)*(-1)^i*(2*n - i)!, i = 0 .. n) end;
# second Maple program:
A:= proc(n, k) A(n, k):= `if`(k=0, n!, A(n+1, k-1) -A(n, k-1)) end:
a:= n-> A(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 22 2019

Mathematica

a[n_] := (2n)!*Hypergeometric1F1[-n, -2n, -1]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 13 2012, after Vladimir Reshetnikov *)

Prog

(Haskell)
a033815 n = a116854 (2 * n + 1) (n + 1)
-- Reinhard Zumkeller, Aug 31 2014

Crossrefs

Main diagonal of array in A068106 and of A047920.

Cf. A000142, A002119, A116854, A267383.

Column k=2 of A372326.

Sequence in context: A236156 A258392 A269504 * A187358 A103916 A201546

Adjacent sequences: A033812 A033813 A033814 * A033816 A033817 A033818

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved