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 A259776 Number A(n,k) of permutations p of [n] with no fixed points and displacement of elements restricted by k: 1 <= |p(i)-i| <= k, square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 4, 0, 0, 1, 0, 1, 2, 9, 6, 1, 0, 1, 0, 1, 2, 9, 24, 13, 0, 0, 1, 0, 1, 2, 9, 44, 57, 24, 1, 0, 1, 0, 1, 2, 9, 44, 168, 140, 45, 0, 0, 1, 0, 1, 2, 9, 44, 265, 536, 376, 84, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,19 COMMENTS Conjecture: Column k > 0 has a linear recurrence (with constant coefficients) of order = A005317(k) = (2^k + C(2*k,k))/2. - Vaclav Kotesovec, Jul 07 2015 From Vaclav Kotesovec, Jul 07 2015: (Start) For k > 1, A(n,k) ~ c(k) * d(k)^n k  c(k)                                  d(k) 2  0.2840509026895102746628049030651...  1.8832035059135258641689474653620... 3  0.1678494211968692989590951622212...  2.6304414743928951523517253855770... 4  0.0973070675347403976445165510589...  3.3758288741377846847522960161445... 5  0.0552389982575367440330445172521...  4.1183824671958029895499633437571... 6  0.0309726120341077011398575643793...  4.8588208495640240252838055706997... 7  0.0172064353582683268003622374813...  5.5979905586951369718393573797927... 8  0.0094902135663231445267663712259...  6.3363450921766600853069060904417... 9  0.00520430877801650454166967632...    7.0741444217884608367707985... 10 0.0028405987031922...                 7.811548995086... (End) LINKS Alois P. Heinz, Antidiagonals n = 0..36, flattened EXAMPLE Square array A(n,k) begins: 1, 1,  1,   1,   1,    1,    1,    1, ... 0, 0,  0,   0,   0,    0,    0,    0, ... 0, 1,  1,   1,   1,    1,    1,    1, ... 0, 0,  2,   2,   2,    2,    2,    2, ... 0, 1,  4,   9,   9,    9,    9,    9, ... 0, 0,  6,  24,  44,   44,   44,   44, ... 0, 1, 13,  57, 168,  265,  265,  265, ... 0, 0, 24, 140, 536, 1280, 1854, 1854, ... MAPLE b:= proc(n, s, k) option remember; `if`(n=0, 1, `if`(n+k in s,       b(n-1, (s minus {n+k}) union `if`(n-k>1, {n-k-1}, {}), k),       add(`if`(j=n, 0, b(n-1, (s minus {j}) union       `if`(n-k>1, {n-k-1}, {}), k)), j=s)))     end: A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, {\$max(1, n-k)..n}, k)): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA b[n_, s_, k_] := b[n, s, k] = If[n==0, 1, If[MemberQ[s, n+k], b[n-1, Join[s ~Complement~ {n+k}] ~Union~ If[n-k>1, {n-k-1}, {}], k], Sum[If[j==n, 0, b[n -1, Join[s ~Complement~ {j}] ~Union~ If[n-k>1, {n-k-1}, {}], k]], {j, s}]] ]; A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Range[Max[1, n-k], n], k]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 29 2017, translated from Maple *) CROSSREFS Columns k=0-10 give: A000007, A059841, A033305, A079997, A259777, A259778, A259779, A259780, A259781, A259782, A259783. Main diagonal gives: A000166. Cf. A259784. Sequence in context: A272728 A174695 A165577 * A116422 A130161 A115672 Adjacent sequences:  A259773 A259774 A259775 * A259777 A259778 A259779 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 05 2015 STATUS approved

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