Search: id:A002493
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%I A002493 M4719 N2017
%S A002493 1,0,0,0,10,60,462,3920,36954,382740,4327510,53088888,702756210,
%T A002493 9988248956,151751644590,2454798429600,42130249479562,764681923900260,
%U A002493 14636063499474054,294639009867223880
%N A002493 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.
%D A002493 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002493 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002493 M. Abramson and W. O. J. Moser, Permutations without rising or falling 
               w-sequences, Ann. Math. Stat., 38 (1967), 1245-1254.
%F A002493 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 
               (zeilberg(AT)math.rutgers.edu), Nov 14 2007
%F A002493 a(n) = Sum((-1)^(n-k)*k!*A102413(n,k),k=1..n), n>2. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Nov 23 2007
%F A002493 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 (vladeta(AT)eunet.rs), Nov 
               24 2007
%p A002493 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 
               (zeilberg(AT)math.rutgers.edu), Nov 14 2007
%Y A002493 Cf. A002464.
%Y A002493 Sequence in context: A112502 A083585 A155633 this_sequence A054364 A004309 
               A052664
%Y A002493 Adjacent sequences: A002490 A002491 A002492 this_sequence A002494 A002495 
               A002496
%K A002493 nonn
%O A002493 1,5
%A A002493 N. J. A. Sloane (njas(AT)research.att.com).

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