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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). 6
1, 1, 14, 426, 24024, 2170680, 287250480, 52370755920, 12585067447680, 3854801333416320, 1465957162768492800, 677696237345719468800, 374281829360322587827200, 243388909697235614324812800, 184070135024053703140543027200 (list; graph; refs; listen; history; text; internal format)
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

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.

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

MAPLE

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

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.

Cf. A002119, A116854, A267383.

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

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Last modified November 21 09:13 EST 2018. Contains 317431 sequences. (Running on oeis4.)