A144084 -id:A144084 - cp's OEIS frontend

A179059 Number of non-attacking placements of 4 rooks on an n X n board.

0, 0, 0, 24, 600, 5400, 29400, 117600, 381024, 1058400, 2613600, 5880600, 12269400, 24048024, 44717400, 79497600, 135945600, 224726400, 360561024, 563376600, 859685400, 1284221400, 1881864600, 2709885024, 3840540000, 5364060000

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Column k=4 of A144084.

Cf. A179058 (3 rooks), A179060 (5 rooks).

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,0,24,600,5400,29400,117600,381024},40] (* Harvey P. Dale, Feb 19 2013 *)
a[n_] := If[n<4, 0, Coefficient[n!*LaguerreL[n, x], x, n-4] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)

a(n) = 4! * binomial(n, 4)^2; \\ Andrew Howroyd, Feb 13 2018

a(n) = 4! * binomial(n, 4)^2.
From Colin Barker, Jan 08 2013: (Start)
a(n) = (n^2*(-6+11*n-6*n^2+n^3)^2)/24.
G.f.: -24*x^4*(x^4 +16*x^3 +36*x^2 +16*x +1) / (x -1)^9.
(End)
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=4} 1/a(n) = (20*Pi^2 - 197)/9.
Sum_{n>=4} (-1)^n/a(n) = (64*log(2) - 44)/9. (End)

A179060 Number of non-attacking placements of 5 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 120, 4320, 52920, 376320, 1905120, 7620480, 25613280, 75271680, 198764280, 480960480, 1082161080, 2289530880, 4594961280, 8809274880, 16225246080, 28844881920, 49689816120, 83217546720, 135870624120

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Column k=5 of A144084.

Cf. A179059 (4 rooks), A179061 (6 rooks).

a[n_] := If[n<5, 0, Coefficient[n!*LaguerreL[n, x], x, n-5] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)

a(n) = 5! * binomial(n, 5)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 5! * binomial(n, 5)^2.

G.f.: -120*x^5*(x+1)*(x^4+24*x^3+76*x^2+24*x+1) / (x-1)^11. - Colin Barker, Jan 08 2013

A179061 Number of non-attacking placements of 6 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 720, 35280, 564480, 5080320, 31752000, 153679680, 614718720, 2120152320, 6492966480, 18036018000, 46172206080, 110279070720, 248127909120, 530024705280, 1081683072000, 2120098821120, 4008311833680

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Column k=5 of A144084.

Cf. A179060 (5 rooks), A179062 (7 rooks).

a(n) = 6! * binomial(n, 6)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 6! * binomial(n, 6)^2.

G.f.: -720*x^6*(x^6+36*x^5+225*x^4+400*x^3+225*x^2+36*x+1) / (x-1)^13. - Colin Barker, Jan 08 2013

A179062 Number of non-attacking placements of 7 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 5040, 322560, 6531840, 72576000, 548856000, 3161410560, 14841066240, 59364264960, 208702494000, 659602944000, 1906252508160, 5104345559040, 12796310741760, 30287126016000, 68146033536000

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Column k=7 of A144084.

Cf. A179061 (6 rooks), A179063 (8 rooks).

7! Binomial[Range[30],7]^2 (* or *) LinearRecurrence[{15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1},{0,0,0,0,0,0,5040,322560,6531840,72576000,548856000,3161410560,14841066240,59364264960,208702494000},30] (* Harvey P. Dale, May 25 2017 *)

a(n) = 7! * binomial(n, 7)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 7!*binomial(n,7)^2.

G.f.: -5040*x^7*(x+1)*(x^6+48*x^5+393*x^4+832*x^3+393*x^2+48*x+1) / (x-1)^15. - Colin Barker, Jan 08 2013

A179063 Number of non-attacking placements of 8 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 40320, 3265920, 81648000, 1097712000, 9879408000, 66784798080, 363606122880, 1669619952000, 6678479808000, 23828156352000, 77203226580480, 230333593351680, 639815537088000, 1669577821632000, 4122835028928000

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Column k=8 of A144084.

Cf. A179062 (7 rooks), A179064 (9 rooks).

a(n) = 8!*binomial(n, 8)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 8!*binomial(n,8)^2.

G.f.: -40320*x^8*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1) / (x -1)^17. - Colin Barker, Jan 08 2013

A179064 Number of non-attacking placements of 9 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 362880, 36288000, 1097712000, 17563392000, 185513328000, 1454424491520, 9090153072000, 47491411968000, 214453407168000, 857813628672000, 3096707199505920, 10237048593408000, 31350961317312000

Offset: 1

Thomas Zaslavsky, Jun 28 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Column k=9 of A144084.

Cf. A179063 (8 rooks), A179065 (10 rooks).

a(n) = 9! * binomial(n, 9)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 9!*binomial(n,9)^2.

G.f.: -362880*x^9*(x +1)*(x^8 +80*x^7 +1216*x^6 +5840*x^5 +10036*x^4 +5840*x^3 +1216*x^2 +80*x +1) / (x -1)^19. - Colin Barker, Jan 08 2013

A179065 Number of non-attacking placements of 10 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 439084800, 15807052800, 296821324800, 3636061228800, 32724551059200, 232707918643200, 1372501805875200, 6948290392243200, 30967071995059200, 123868287980236800

Offset: 1

Thomas Zaslavsky, Jun 28 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Column k=10 of A144084.

Cf. A179064 (9 rooks).

a(n) = 10! * binomial(n, 10)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 10! * binomial(n, 10)^2.

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880

Offset: 0

Peter Luschny, Mar 28 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,   2;
  [3] 0, 3,  12,     6;
  [4] 0, 4,  36,    72,     24;
  [5] 0, 5,  80,   360,    480,     120;
  [6] 0, 6, 150,  1200,   3600,    3600,     720;
  [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
  [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
  [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.

Cf. A144084, A021010.

A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n,k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board.

Original entry on oeis.org

0, 1, 4, 13, 31, 66, 123, 214, 346, 535, 790, 1131, 1569, 2128, 2821, 3676, 4708, 5949, 7416, 9145, 11155, 13486, 16159, 19218, 22686, 26611, 31018, 35959, 41461, 47580, 54345, 61816, 70024, 79033, 88876, 99621, 111303, 123994, 137731, 152590, 168610, 185871

Offset: 1

Mo Li, Apr 19 2019

			For n = 4 the a(4) = 13 solutions are
{{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
{{0,1,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
—————————————————————————————————————
{{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,1}}
—————————————————————————————————————
{{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
{{1,0,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
—————————————————————————————————————
{{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,1,0}}
—————————————————————————————————————
{{0,0,0,0}}
{{0,1,0,0}}
{{0,0,1,0}}
{{0,0,0,0}}

Leisure Maths Entertainment Forum, 2 nonattacking rooks on n X n board, Chinese blog.

Cf. A000903, A163102, A035287, A179058, A144084.

Table[
Piecewise[{{(n (n^3 - 2 n^2 + 6 n - 4))/16, Mod[n, 2] == 0},
{((n - 1) (n^3 - n^2 + 5 n - 1))/16, Mod[n, 2] == 1}}],{n, 20}]

a(n) = (1/16)*n*(n^3-2n^2+6n-4) if n is even;
a(n) = (1/16)*(n-1)*(n^3-n^2+5n-1) if n is odd.
G.f.: -x^2*(x^2+1)*(x^2+x+1)/((x+1)^2*(x-1)^5). - Alois P. Heinz, Apr 26 2019

A373766 Triangle read by rows, T(n,k) with n,k > 1, is the number of subsequences of length k over all permutations of [n], which are neither increasing or decreasing.

Original entry on oeis.org

0, 0, 4, 0, 64, 22, 0, 800, 550, 118, 0, 9600, 9900, 4248, 718, 0, 117600, 161700, 104076, 35182, 5038, 0, 1505280, 2587200, 2220288, 1125824, 322432, 40318, 0, 20321280, 41912640, 44960832, 30397248, 13058496, 3265758, 362878, 0, 290304000, 698544000, 899216640, 759931200, 435283200

Offset: 2

Thomas Scheuerle, Jun 18 2024

Column T(n, 1) was omitted in the presentation of this sequence. Its definition may depend on the usage. In the combinatorics of subsequences it may be convenient to define T(n, 1) = 0, but if this sequence will be interpreted differently for example as polynomial coefficients then T(n, 1) = -n*n! could be a mathematically more natural definition.

			The triangle begins:
 n| k: 2|      3|      4|      5|     6|    7|
==============================================
[2]    0,
[3]    0,      4
[4]    0,     64,     22
[5]    0,    800,    550,    118
[6]    0,   9600,   9900,   4248,   718
[7]    0, 117600, 161700, 104076, 35182, 5038
.
T(3, 3) = 4 because:
  {1, 2, 3} has no subsequences which are neither increasing or decreasing.
  {1, 3, 2} has {1, 3, 2}
  {2, 1, 3} has {2, 1, 3}
  {2, 3, 1} has {2, 3, 1}
  {3, 1, 2} has {3, 1, 2}
  {3, 2, 1} has no subsequences which are neither increasing or decreasing.

Cf. A144084.

T(n, k) = n!*binomial(n, k)-2*((n-k)! * binomial(n, n-k)^2)

row(n) = if(n==2, [0], abs(Vecrev(-n!*((-1)^n*2*pollaguerre(n)-(-1+x)^n))[3..n+1]))

T(n, k) = n!*binomial(n, k)-2*((n - k)! * binomial(n, n - k)^2).
An alternative definition of T(n, k) which includes k < 2 can be done by Laguerre polynomials:
 Sum_{k=0..n} T(n, k)*x^k = n!*((1 + x)^n - 2*L_{n}(-x)), where L_{n} is the n-th Laguerre polynomial.

cp's OEIS Frontend

A179059 Number of non-attacking placements of 4 rooks on an n X n board.

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A179060 Number of non-attacking placements of 5 rooks on an n X n board.

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A179061 Number of non-attacking placements of 6 rooks on an n X n board.

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A179062 Number of non-attacking placements of 7 rooks on an n X n board.

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A179063 Number of non-attacking placements of 8 rooks on an n X n board.

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A179064 Number of non-attacking placements of 9 rooks on an n X n board.

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A179065 Number of non-attacking placements of 10 rooks on an n X n board.

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A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

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A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board.

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