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A002493
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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.
(Formerly M4719 N2017)
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2
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1, 0, 0, 0, 10, 60, 462, 3920, 36954, 382740, 4327510, 53088888, 702756210, 9988248956, 151751644590, 2454798429600, 42130249479562, 764681923900260, 14636063499474054, 294639009867223880
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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REFERENCES
| M. Abramson and W. O. J. Moser, Permutations without rising or falling w-sequences, Ann. Math. Stat., 38 (1967), 1245-1254.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| 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
a(n) = Sum((-1)^(n-k)*k!*A102413(n,k),k=1..n), n>2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 23 2007
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
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MAPLE
| 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
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CROSSREFS
| Cf. A002464.
Sequence in context: A112502 A083585 A155633 * A054364 A004309 A052664
Adjacent sequences: A002490 A002491 A002492 * A002494 A002495 A002496
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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