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

1,2

Links

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

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

Example

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).

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 *)

Prog

(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

Crossrefs

Cf. A095236, A192008, A192009, A276657.

Sequence in context: A335111 A337882 A095239 * A065953 A371832 A118211

Adjacent sequences: A192006 A192007 A192008 * A192010 A192011 A192012

Keyword

nonn

Author

Jens Voß, Jun 21 2011

Extensions

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

Status

approved