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%I #33 Jan 16 2023 14:56:26
%S 0,0,2,12,96,720,6480,60480,645120,7257600,90720000,1197504000,
%T 17244057600,261534873600,4271736268800,73229764608000,
%U 1339058552832000,25609494822912000,518592270163968000,10948059036794880000,243290200817664000000
%N a(n) = floor(n^2/4)*n!.
%C a(n) is the total displacement of all letters in all permutations on n letters as if the first letter were connected to the last letter, forming a loop.
%C For the sequence A090672 the displacement of the permutation "0123" is 0, while that of the permutation "3210" is 8 because each of the digits 0 and 3 is 3 places away from its original place and each of the digits 1 and 2 is one place away, so the total displacement is 3+1+1+3 = 8.
%C In this sequence, however, the displacement is calculated differently: that of "0123" is 0 as before, but the displacement of "3210" is no longer 8 because the first index and last index are connected, forming a loop; each of the digits 0 and 3 is now 1 place away from its original place (and each of the digits 1 and 2 is one place away, as before), so the total displacement is calculated as 1+1+1+1 = 4.
%H Alaa Sultan Al-hasani, <a href="/A306258/b306258.txt">Table of n, a(n) for n = 0..446</a>
%F a(n) = floor(n^2/4)*n!.
%F a(n) = A002620(n)*n!.
%F a(n) = A077613(n)*n.
%F E.g.f.: x^2/((x+1)*(1-x)^3). - _Alois P. Heinz_, Feb 01 2019
%t Table[Floor[n^2/4]n!,{n,0,40}] (* _Harvey P. Dale_, Jan 16 2023 *)
%o (PARI) a(n) = floor(n^2/4)*n!;
%Y Cf. A002620, A077613.
%K nonn
%O 0,3
%A _Alaa Sultan Al-hasani_, Feb 01 2019
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