A192009 Modified cyclic phone booth sequence: number of ways to occupy n labeled phone booths in a circle one by one, each time picking a phone booth adjacent to the smallest number of previously occupied phone booths.

1, 2, 6, 8, 40, 168, 504, 3456, 15552, 97920, 620928, 4465152, 31449600, 273369600, 2172096000, 20968243200, 192753561600, 2032260710400, 20942298316800, 243270107136000, 2758764950323200, 34958441123020800, 434690126954496000, 5946571752210432000, 80503989505228800000

Offset: 1

Author

Jens Voß, Jun 21 2011

Keywords

• nonn

Examples

			For n=4, the A192009(n) = 6 ways of picking the phone booths are (1, 3, 2, 4), (1, 3, 4, 2), (2, 4, 1, 3), (2, 4, 3, 1), (3, 1, 2, 4), (3, 1, 4, 2), (4, 2, 1, 3), (4, 2, 3, 1).
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A095236, A192008, A192009, A276657.

Programs

Maple

A192009 := proc(n)
    local a,k,m;
    if n = 1 then
        return 1;
    end if;
    a := 0 ;
    for k from 0 to n/3 do
        m := (n-3*k)/2 ;
        if type (m,'integer') then
            a := a+(m+k-1)!*binomial(m+k,m)*2^k*k!*(m+k)! ;
        end if;
    end do:
    a*n ;
end proc:
seq(A192009(n),n=1..20) ; # R. J. Mathar, Sep 17 2016

Mathematica

r[n_] := {ToRules[Reduce[m >= 0 && k >= 0 && 2m+3k == n, {m, k}, Integers] ]}; f[{m_, k_}] := (m+k-1)!*Binomial[m + k, m]*2^k*k!*(m+k)!; a[n_] := n*Total[f /@ ({m, k} /. r[n])]; a[1] = 1; Array[a, 25] (* Jean-François Alcover, Sep 13 2016, after Max Alekseyev *)

PARI

{ A192009(n) = my(r,k); if(n==1,return(1)); r=0; forstep(m=lift(Mod(n,3)/2),n\2,3, k=(n-2*m)\3; r+=(m+k-1)!*binomial(m+k,m)*2^k*k!*(m+k)!); r*n; } \\ Max Alekseyev, Sep 11 2016

Formula

For n > 1, a(n) = n * Sum (m+k-1)!*binomial(m+k,m)*2^k*k!*(m+k)!, where the sum is taken over nonnegative m,k such that 2*m+3*k = n. - Max Alekseyev, Sep 11 2016

a(n) = n * A276657(n). - Max Alekseyev, Sep 11 2016

Extensions

Terms a(15) onward from Max Alekseyev, Sep 11 2016