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