A306258 a(n) = floor(n^2/4)*n!.

0, 0, 2, 12, 96, 720, 6480, 60480, 645120, 7257600, 90720000, 1197504000, 17244057600, 261534873600, 4271736268800, 73229764608000, 1339058552832000, 25609494822912000, 518592270163968000, 10948059036794880000, 243290200817664000000

Offset: 0

Author

Alaa Sultan Al-hasani, Feb 01 2019

Keywords

• nonn

Comments

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.

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.

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.

Links

• Alaa Sultan Al-hasani, Table of n, a(n) for n = 0..446

Crossrefs

Cf. A002620, A077613.

Programs

Mathematica

Table[Floor[n^2/4]n!,{n,0,40}] (* Harvey P. Dale, Jan 16 2023 *)

PARI

a(n) = floor(n^2/4)*n!;

Formula

a(n) = floor(n^2/4)*n!.

a(n) = A002620(n)*n!.

a(n) = A077613(n)*n.

E.g.f.: x^2/((x+1)*(1-x)^3). - Alois P. Heinz, Feb 01 2019