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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved