OFFSET
0,3
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
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
MATHEMATICA
Table[Floor[n^2/4]n!, {n, 0, 40}] (* Harvey P. Dale, Jan 16 2023 *)
PROG
(PARI) a(n) = floor(n^2/4)*n!;
CROSSREFS
KEYWORD
nonn
AUTHOR
Alaa Sultan Al-hasani, Feb 01 2019
STATUS
approved