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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.

%I A192009 #24 Sep 29 2019 02:37:51
%S A192009 1,2,6,8,40,168,504,3456,15552,97920,620928,4465152,31449600,
%T A192009 273369600,2172096000,20968243200,192753561600,2032260710400,
%U A192009 20942298316800,243270107136000,2758764950323200,34958441123020800,434690126954496000,5946571752210432000,80503989505228800000
%N 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.
%H A192009 Max Alekseyev, <a href="/A192009/b192009.txt">Table of n, a(n) for n = 1..100</a>
%F A192009 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
%F A192009 a(n) = n * A276657(n). - _Max Alekseyev_, Sep 11 2016
%e A192009 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).
%p A192009 A192009 := proc(n)
%p A192009     local a,k,m;
%p A192009     if n = 1 then
%p A192009         return 1;
%p A192009     end if;
%p A192009     a := 0 ;
%p A192009     for k from 0 to n/3 do
%p A192009         m := (n-3*k)/2 ;
%p A192009         if type (m,'integer') then
%p A192009             a := a+(m+k-1)!*binomial(m+k,m)*2^k*k!*(m+k)! ;
%p A192009         end if;
%p A192009     end do:
%p A192009     a*n ;
%p A192009 end proc:
%p A192009 seq(A192009(n),n=1..20) ; # _R. J. Mathar_, Sep 17 2016
%t A192009 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_ *)
%o A192009 (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
%Y A192009 Cf. A095236, A192008, A192009, A276657.
%K A192009 nonn
%O A192009 1,2
%A A192009 _Jens Voß_, Jun 21 2011
%E A192009 Terms a(15) onward from _Max Alekseyev_, Sep 11 2016