cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A183232 Second of two complementary trees generated by the triangular numbers. The other tree is A183231.

Original entry on oeis.org

2, 8, 5, 53, 12, 26, 9, 1538, 63, 103, 17, 404, 33, 64, 14, 1186568, 1593, 2143, 74, 5563, 117, 188, 23, 82619, 432, 628, 41, 2209, 75, 134, 20, 703974775733, 1188108, 1272808, 1649, 2301583, 2208, 2924, 86, 15487393
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2011

Keywords

Comments

See A183231 (first tree).

Examples

			First 3 levels:
....................2
...............8...........5
............53...12.....26...9

Crossrefs

Cf. A183079, A183231, A183244.

Formula

See the formulas at A183231 and A183244.

A183245 Number of permutations of 1..2*n+2 with each element displaced by at least n.

Original entry on oeis.org

9, 29, 112, 436, 1708, 6724, 26572, 105316, 418348, 1664644, 6632332, 26450596, 105566188, 421556164, 1684098892, 6730018276, 26900941228, 107546369284, 430013290252, 1719536600356, 6876596719468, 27501737832004, 109993004190412
Offset: 1

Views

Author

R. H. Hardin, Jan 03 2011

Keywords

Comments

Row 3 of A183244.

Examples

			All permutations of 1-6 with minimum displacement 2:
(4,5,1,6,2,3) (4,5,1,6,3,2) (4,5,6,1,2,3) (4,5,6,1,3,2) (4,5,6,2,1,3)
(4,5,6,2,3,1) (4,6,5,1,2,3) (4,6,5,1,3,2) (4,6,5,2,1,3) (4,6,5,2,3,1)
(5,4,1,6,2,3) (5,4,1,6,3,2) (5,4,6,1,2,3) (5,4,6,1,3,2) (5,4,6,2,1,3)
(5,4,6,2,3,1) (5,6,1,2,3,4) (3,4,5,6,1,2) (3,4,5,6,2,1) (3,5,1,6,2,4)
(3,5,6,1,2,4) (3,5,6,2,1,4) (3,6,5,1,2,4) (3,6,5,2,1,4) (6,4,5,1,2,3)
(6,4,5,1,3,2) (6,4,5,2,1,3) (6,4,5,2,3,1) (6,5,1,2,3,4)

Links

Crossrefs

Cf. A183244.

Formula

Empirical (for n>=2): 25*4^(n-2) + 4*3^(n-2). - Vaclav Kotesovec, Nov 27 2012
Conjectures from Colin Barker, Mar 27 2018: (Start)
G.f.: x*(9 - 34*x + 17*x^2) / ((1 - 3*x)*(1 - 4*x)).
a(n) = 7*a(n-1) - 12*a(n-2) for n>3.
(End)

A183246 Number of permutations of 1..2*n+3 with each element displaced by at least n.

Original entry on oeis.org

44, 206, 1168, 6984, 41808, 250464, 1501248, 9001344, 53983488, 323802624, 1942422528, 11652962304, 69911482368, 419443728384, 2516561707008, 15098967588864, 90592194920448, 543546727071744, 3261254592626688
Offset: 1

Views

Author

R. H. Hardin, Jan 03 2011

Keywords

Comments

Row 4 of A183244.

Examples

			Some permutations of 1-7 with minimum displacement 2:
(4,5,6,1,7,3,2) (3,6,5,7,2,4,1) (7,6,1,2,3,4,5) (6,4,1,2,7,3,5) (5,6,7,1,3,2,4)
(3,7,5,6,1,4,2) (5,6,1,2,7,3,4) (6,5,7,2,1,4,3) (6,5,1,7,3,4,2) (6,5,1,7,2,4,3)
(7,5,6,1,2,3,4) (5,7,1,6,2,3,4) (3,4,6,7,1,2,5) (4,6,5,7,2,1,3) (7,4,5,6,1,2,3)
(6,4,7,2,1,3,5) (4,6,1,7,3,2,5) (5,6,1,2,7,4,3) (5,4,6,7,1,3,2) (3,5,7,6,2,1,4)

Links

Crossrefs

Cf. A183244.

Formula

Empirical (for n>=3): 289/9*6^(n-1) + 3*4^(n-2). - Vaclav Kotesovec, Nov 27 2012
Conjectures from Colin Barker, Mar 27 2018: (Start)
G.f.: 2*x*(22 - 117*x + 82*x^2 + 124*x^3) / ((1 - 4*x)*(1 - 6*x)).
a(n) = 10*a(n-1) - 24*a(n-2) for n>4.
(End)

A183247 Number of permutations of 1..2*n+4 with each element displaced by at least n.

Original entry on oeis.org

265, 1708, 13365, 114124, 998112, 8751552, 76915200, 677461056, 5979015552, 52866428352, 468241962240, 4153739516736, 36900195503232, 328234303349952, 2923170205716480, 26061029280983616, 232569271090134912
Offset: 1

Views

Author

R. H. Hardin, Jan 03 2011

Keywords

Comments

Row 5 of A183244.

Examples

			Some permutations of 1-8 with minimum displacement 2:
(7,4,5,8,3,2,1,6) (8,6,5,7,1,3,4,2) (8,4,5,6,7,2,1,3) (3,7,1,6,8,4,2,5)
(8,4,6,2,7,3,5,1) (6,5,7,8,3,2,1,4) (3,7,6,8,2,1,5,4) (5,4,6,7,8,1,2,3)
(4,8,6,7,2,1,3,5) (6,7,5,8,2,1,4,3) (7,5,6,1,8,3,2,4) (3,8,5,2,7,1,4,6)
(4,7,1,2,8,3,5,6) (7,4,5,1,8,2,3,6) (7,6,5,2,8,3,4,1) (3,5,7,8,1,4,2,6)
(5,6,7,8,2,4,3,1) (7,5,6,8,2,1,3,4) (5,8,6,7,2,3,1,4) (3,6,8,1,7,4,2,5)

Links

Crossrefs

Cf. A183244.

Formula

Empirical (for n>=4): 85264*9^(n-4) + 7203*2^(3*n-10) + 48*5^(n-4). - Vaclav Kotesovec, Nov 27 2012
Conjectures from Colin Barker, Mar 27 2018: (Start)
G.f.: x*(265 - 4122*x + 17394*x^2 - 7150*x^3 - 29191*x^4 - 100844*x^5) / ((1 - 5*x)*(1 - 8*x)*(1 - 9*x)).
a(n) = 22*a(n-1) - 157*a(n-2) + 360*a(n-3) for n>6.
(End)

A306543 Number T(n,k) of permutations p of [n] such that |p(j)-j| >= k (for all j in [n]); triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 9, 1, 120, 44, 4, 720, 265, 29, 1, 5040, 1854, 206, 8, 40320, 14833, 1708, 112, 1, 362880, 133496, 15702, 1168, 16, 3628800, 1334961, 159737, 13365, 436, 1, 39916800, 14684570, 1780696, 159414, 6984, 32, 479001600, 176214841, 21599745, 2036488, 114124, 1708, 1
Offset: 0

Views

Author

Alois P. Heinz, Feb 22 2019

Keywords

Examples

			Triangle T(n,k) begins:
          1;
          1;
          2,         1;
          6,         2;
         24,         9,        1;
        120,        44,        4;
        720,       265,       29,       1;
       5040,      1854,      206,       8;
      40320,     14833,     1708,     112,      1;
     362880,    133496,    15702,    1168,     16;
    3628800,   1334961,   159737,   13365,    436,    1;
   39916800,  14684570,  1780696,  159414,   6984,   32;
  479001600, 176214841, 21599745, 2036488, 114124, 1708, 1;
  ...

Links

Crossrefs

Columns k=0-6 give (offsets may differ): A000142, A000166, A001883, A075851, A075852, A183242, A183243.
T(2n,n) gives A000012.
T(2n+1,n) gives A000079.
T(2n+2,n) gives A183245 for n > 0.
T(2n+3,n) gives A183246 for n > 0.
T(2n+4,n) gives A183247 for n > 0.
Cf. A183244, A299789.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(n=0, 1, LinearAlgebra[
      Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))
    end:
seq(seq(T(n, k), k=0..floor(n/2)), n=0..12);
  • Mathematica
    T[n_, k_] := T[n, k] = If[n==0, 1, Permanent[Table[
     If[Abs[i-j] >= k, 1, 0], {i, n}, {j, n}]]];
Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 26 2021, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{j=k..floor(n/2)} A299789(n,j) for n > 0.

A183242 Number of permutations of 1..n+9 with each element displaced by at least 5.

Original entry on oeis.org

1, 32, 1708, 41808, 998112, 21201024, 441629332, 9154333160, 192565379941, 4146526612518, 91932770123800, 2105115145527440, 49865459492032640, 1222725452864637888, 31038395731257896576, 815438320994063139856
Offset: 1

Views

Author

R. H. Hardin, Jan 03 2011

Keywords

Comments

a(n) equals the permanent of the (n+9) X (n+9) matrix with 0's along the main diagonal and the next 4 superdiagonals and subdiagonals, and 1's everywhere else. - John M. Campbell, Jul 09 2011

Examples

			Some permutations of 1-13 with minimum displacement 5:
(8,13,9,12,10,11,1,2,3,4,5,7,6) (6,11,8,9,10,13,12,1,2,5,3,7,4)
(9,7,8,11,10,1,12,13,3,5,2,4,6) (7,8,10,9,13,11,12,2,1,3,5,6,4)
(8,7,9,10,11,13,12,3,1,5,4,2,6) (7,9,8,10,11,13,12,1,4,2,6,5,3)
(6,12,8,9,10,11,2,13,3,4,1,5,7) (9,7,11,10,13,12,2,1,3,4,5,6,8)
(6,12,8,9,10,11,13,2,1,4,5,7,3) (7,9,10,11,13,12,2,3,1,4,6,5,8)
(6,9,8,10,11,12,13,2,4,3,1,7,5) (10,7,8,9,11,12,13,2,1,5,3,4,6)
(6,8,9,10,12,11,2,13,3,1,4,7,5) (9,8,12,13,10,11,2,1,4,5,3,7,6)
(8,12,9,13,10,11,2,1,3,5,4,7,6) (7,9,8,10,12,11,13,3,1,5,2,4,6)
(10,9,8,11,13,12,1,3,2,5,4,7,6) (7,8,9,11,10,1,12,13,3,5,6,4,2)
(7,8,13,9,10,11,12,1,4,5,2,6,3) (8,10,11,9,13,1,12,2,4,3,6,7,5)

Crossrefs

Column 5 of A183244.

A183243 Number of permutations of 1..n+11 with each element displaced by at least 6.

Original entry on oeis.org

1, 64, 6724, 250464, 8751552, 252813312, 6860776320, 178195229760, 4564491262444, 116967725946488, 3030418865421197, 79901439039073238, 2153387935521461560, 59484865107386246288
Offset: 1

Views

Author

R. H. Hardin, Jan 03 2011

Keywords

Comments

a(n) equals the permanent of the (n+11) X (n+11) matrix with 0's along the main diagonal, the first 5 superdiagonals and subdiagonals, and 1's everywhere else. - John M. Campbell, Jul 09 2011

Examples

			Some permutations of 1-15 with minimum displacement 6:
(12,9,15,10,11,13,14,1,3,4,5,6,7,2,8) (10,9,11,14,12,13,15,1,3,4,2,6,5,7,8)
(8,12,15,10,11,13,14,1,2,3,4,6,7,5,9) (12,11,9,10,15,13,1,14,2,3,5,4,7,6,8)
(12,8,9,10,11,14,13,2,15,1,5,6,7,3,4) (12,8,11,10,15,13,1,14,2,3,4,6,7,5,9)
(8,11,9,10,12,13,14,1,15,2,3,5,6,4,7) (10,8,9,12,11,13,15,14,1,2,3,5,7,4,6)
(10,9,12,11,13,15,1,14,2,3,4,5,7,6,8) (8,15,11,10,12,13,14,1,2,3,4,5,6,7,9)
(12,10,9,11,15,13,14,2,3,4,5,1,6,8,7) (12,8,9,10,11,13,14,1,15,3,4,5,7,2,6)
(8,14,9,10,11,12,13,15,1,3,2,5,6,4,7) (10,9,11,12,13,14,1,2,15,4,5,6,3,7,8)
(12,9,15,10,11,13,14,2,3,1,4,6,5,8,7) (10,9,15,12,11,14,13,2,3,1,4,6,7,8,5)
(8,10,9,13,11,12,1,14,15,2,4,3,6,7,5) (12,10,9,13,11,15,14,1,2,4,3,5,6,7,8)
(10,8,9,11,13,12,15,14,2,1,4,3,5,7,6) (8,10,12,11,13,15,14,1,2,4,5,3,6,7,9)

Crossrefs

Column 6 of A183244.

Programs

  • Maple
    with(LinearAlgebra): nmax:=10: for k from 1 to nmax do T:=Matrix(1..k+11,1..k+11,1); for i from 0 to 5 do for n from 1 to k+11-i do T[n,n+i]:=0 od: od: for j from 1 to 5 do for n from j+1 to k+11 do T[n,n-j]:= 0: od: od: a(k):= Permanent(T); od: seq(a(n),n=1..nmax); # Johannes W. Meijer, Jul 09 2011
  • Mathematica
    a[n_] := Permanent[Table[If[Abs[i-j]<6, 0, 1], {i, 1, n+11}, {j, 1, n+11}] ]; Table[an = a[n]; Print["a(", n , ") = ", an]; an, {n, 1, 14}] (* Jean-François Alcover, Jan 07 2016 *)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.