|
%I #29 Jan 11 2024 08:58:20
%S 1,6,30,164,1030,7422,60620,554248,5611770,62353010,754471432,
%T 9876716940,139097096918,2097156230470,33704296561140,575219994643472,
%U 10389911153247730,198019483156015578,3971390745517868000,83608226221428800020,1843561388182505040462
%N Number of tilings of an n-cell circular array with rectangular tiles of any size, and where the number of possible colors of a tile is given by the largest cell covered.
%H Jonathan Beagley and Lara Pudwell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Pudwell/pudwell13.html">Colorful Tilings and Permutations</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
%F a(n) = n * Sum_{k=1..n} n!/k!.
%F a(n) = n * A002627(n).
%F From _Alois P. Heinz_, Jun 28 2021: (Start)
%F E.g.f.: (exp(x)-x)/(x-1)^2 - exp(x).
%F a(n) = A193657(n) - 1. (End)
%F D-finite with recurrence a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2) -2 =0. - _R. J. Mathar_, Jan 11 2024
%p a:= proc(n) a(n):= `if`(n=1, 1, a(n-1)*n^2/(n-1)+n) end:
%p seq(a(n), n=1..21); # _Alois P. Heinz_, Jun 28 2021
%t With[{r = Range[21]}, r*Rest@ FoldList[Times @@ {##} + 1 &, 0, r]] (* _Michael De Vlieger_, Jun 28 2021 *)
%o (PARI) a(n) = n*sum(k=1, n, n!/k!); \\ _Michel Marcus_, Jun 29 2021
%Y Cf. A002627, A193657.
%K nonn
%O 1,2
%A _Lara Pudwell_, Jun 28 2021
|
