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%I M4719 N2017 #73 Dec 19 2021 10:01:42
%S 1,0,0,0,10,60,462,3920,36954,382740,4327510,53088888,702756210,
%T 9988248956,151751644590,2454798429600,42130249479562,764681923900260,
%U 14636063499474054,294639009867223880
%N Number of ways to arrange n non-attacking kings on an n X n board, with 2 sides identified to form a cylinder, with 1 in each row and column.
%C Number of directed Hamiltonian paths in the complement of C_n where C_n is the n-cycle graph. - _Andrew Howroyd_, Mar 15 2016
%C Number of ways of arranging n consecutive integers in a circle such that no pair of adjacent integers differ by 1, rotations are distinct. - _Graham Holmes_, Sep 03 2020
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002493/b002493.txt">Table of n, a(n) for n = 1..100</a>
%H M. Abramson and W. O. J. Moser, <a href="http://dx.doi.org/10.1214/aoms/1177698793">Permutations without rising or falling w-sequences</a>, Ann. Math. Stat., 38 (1967), 1245-1254.
%H Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, <a href="https://arxiv.org/abs/2001.02948">Counting King Permutations on the Cylinder</a>, arXiv:2001.02948 [math.CO], 2020.
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 627-8.
%H A. J. Schwenk, <a href="/A002464/a002464_1.pdf">Letter to N. J. A. Sloane, Mar 24 1980</a> (with enclosure and notes)
%F The linear recurrence operator annihilating this sequence is (N is the shift operator Na(n):=a(n + 1)) is - 3*(43*n + 197)*(n - 2)*(n + 1)/( - 1222 + 753*n + 349*n^2) - 5*(n - 1)*(44*n^2 + 477*n + 1222)/( - 1222 + 753*n + 349*n^2)*N + 2*(n + 1)*(239*n^2 + 873*n - 1232)/( - 1222 + 753*n + 349*n^2)*N^2 + 4*(394 - 259*n + 215*n^2 + 55*n^3)/( - 1222 + 753*n + 349*n^2)*N^3 - ( - 7342 + 3699*n + 2718*n^2 + 349*n^3)/( - 1222 + 753*n + 349*n^2)*N^4 + N^5. - _Doron Zeilberger_, Nov 14 2007
%F a(n) = Sum((-1)^(n-k)*k!*A102413(n,k),k=1..n), n>2. - _Vladeta Jovovic_, Nov 23 2007
%F a(n) = b(n+1) - 2*Sum_{k=0..floor(n/2)} b(n-2*k) for n>1, where b(n)=A002464(n) if n>0 else b(0)=0. - _Vladeta Jovovic_, Nov 24 2007
%F Asymptotic: a(n) ~ n!/e^2*(1 - 2/n - 2/n^2 - 4/(3n^3) + 8/(3n^4) + 326/(15n^5) + 4834/(45n^6) + 154258/(315n^7) + 232564/(105n^8) + ...). - _Vaclav Kotesovec_, Apr 06 2012
%F a(n) = n! + Sum_{i=1..n-1} ((-1)^i * (n-i-1)! * n * Sum_{j=0..i-1} (2^(j+1) * C(i-1,j) * C(n-i,j+1))), for n>=5. - _Andrew Woods_, Jan 08 2015
%p b1:= proc(n, r) local gu, x; if r=0 then RETURN(0): fi: gu := (x*diff(x*(1+x)/(1-x),x))* (x*(1 + x)/(1 - x))^(r-1); gu := taylor(gu, x = 0, n +1); coeff(gu, x, n ) end: b:=proc(n) local r: if n=1 then 1 elif n=2 then 0 else add((-1)^(n-r)*r!*b1(n,r),r=0..n): fi: end: # _Doron Zeilberger_, Nov 14 2007
%t b[n_]:=(If[n>0, n!+Sum[(-1)^r*(n-r)!*Sum[2^c*Binomial[r-1, c-1]*Binomial[n-r,c], {c, 1, r}], {r, 1, n-1}], 0]); Table[If[n>2, b[n]-2*Sum[b[n-1-2k], {k, 0, Floor[n/2]}], If[n==1, 1, 0]], {n, 1, 25}] (* _Vaclav Kotesovec_ after _Vladeta Jovovic_, Apr 06 2012 *)
%Y Cf. A002464, A002816, A006184.
%Y Right diagonal of A338838.
%K nonn
%O 1,5
%A _N. J. A. Sloane_
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