A260575 -id:A260575 - cp's OEIS frontend

A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.

1, 11, 72, 367, 1630, 6680, 26082, 98870, 368045, 1354850, 4953503, 18035279, 65499031, 237511321, 860471110, 3115667369, 11277816388, 40814611818, 147692103728, 534404499040, 1933597628291, 6996040095316, 25312367524557, 91581960107817, 331348634005165

Offset: 2

Jeffrey Davis, Jul 29 2015

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 2 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 2..1000
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (12,-59,155,-236,209,-100,20).

Column k=2 of A269742.

Cf. A260575, A260582, A260583, A260584, A260727.

CoefficientList[Series[-(5*x^4 - 3*x^3 - x^2 - x + 1)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 16 2015 *)

Vec(-(5*x^6 - 3*x^5 - x^4 - x^3 + x^2)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1) + O(x^40)) \\ Michel Marcus, Aug 17 2015

G.f.: -x^2*(5*x^4-3*x^3-x^2-x+1)/((1-5*x+5*x^2)*(2*x-1)^2*(x-1)^3).
a(n) = 12*a(n-1) - 59*a(n-2) + 155*a(n-3) - 236*a(n-4) + 209*a(n-5) - 100*a(n-6) + 20*a(n-7). - Wesley Ivan Hurt, Jan 01 2024
a(n) = (n+2)*(n-1)/2-2^n*(1+3*n/2)+2*A030191(n)-5*A030191(n-1). - R. J. Mathar, Aug 26 2025

A260582 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

Original entry on oeis.org

1, 72, 2438, 48965, 727982, 9002669, 98831244, 1001534339, 9604385112, 88600727292, 795108048465, 6995452987296, 60672964077315, 520801298224219, 4436874672072459, 37592602817393616, 317246106027904761, 2669508900483670024, 22415690107381454687

Offset: 3

Scarlitte Ponce, Jul 29 2015

nonn

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 4 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that the number of throws is between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 3..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Column k=4 of A269742.

Cf. A260575, A260583, A260584, A260585, A260727.

CoefficientList[Series[-(4584825 x^18 - 32402639 x^17 + 116197885 x^16 - 276109240 x^15 + 466638686 x^14 - 565360388 x^13 + 479478061 x^12 - 263101580 x^11 + 64737485 x^10 + 27721713 x^9 - 36043190 x^8 + 18319939 x^7 - 5637417 x^6 + 1094626 x^5 - 124221 x^4 + 5875 x^3 + 171 x^2 - 14 x - 1)/(19266000 x^22 - 234624000 x^21 + 1345258600 x^20 - 4829459800 x^19 + 12177772645 x^18 - 22934336190  x^17 + 33487611783 x^16 - 38844575208 x^15 + 36384232939 x^14 - 27820436326 x^13 + 17485343731 x^12 - 9066508172 x^11 + 3881838842 x^10 - 1369857572 x^9 + 396588486 x^8 - 93458208 x^7 + 17715207 x^6 - 2654590 x^5 + 306583 x^4 - 26260 x^3 + 1567 x^2 - 58 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 19 2015 *)

G.f.: -(4584825*x^21 - 32402639*x^20 + 116197885*x^19 - 276109240*x^18 + 466638686*x^17 - 565360388*x^16 + 479478061*x^15 - 263101580*x^14 + 64737485*x^13 + 27721713*x^12 - 36043190*x^11 + 18319939*x^10 - 5637417*x^9 + 1094626*x^8 - 124221*x^7 + 5875*x^6 + 171*x^5 - 14*x^4 - x^3)/(19266000*x^22 - 234624000*x^21 + 1345258600*x^20 - 4829459800*x^19 + 12177772645*x^18 - 22934336190*x^17 + 33487611783*x^16 - 38844575208*x^15 + 36384232939*x^14 - 27820436326*x^13 + 17485343731*x^12 - 9066508172*x^11 + 3881838842*x^10 - 1369857572*x^9 + 396588486*x^8 - 93458208*x^7 + 17715207*x^6 - 2654590*x^5 + 306583*x^4 - 26260*x^3 + 1567*x^2 - 58*x + 1).

A260583 Number of ways to place 3n rooks on an n X n board, 3 rooks in each row and each column, multiple rooks in an allowed cell, and exactly 4 rooks below the main diagonal.

Original entry on oeis.org

11, 595, 14679, 253247, 3564803, 44226950, 505572550, 5473391465, 57122380066, 581477852342, 5819301681925, 57564437594318, 564911137682637, 5513703983635512, 53616132982114742, 520057429817203110, 5035740328012627416, 48704838658567681135

Offset: 3

Esther Banaian, Jul 29 2015

nonn

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 4 balls when we can catch/throw up to 3 balls at a time. (Minimal in the sense that the throw heights are between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 3..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Column k=4 of A269743.

Cf. A260575, A260582, A260584, A260585, A260727.

G.f.: -(24558000*x^24 - 221169800*x^23 + 1030045255*x^22 - 3270869391*x^21 + 7705144467*x^20 - 13843184523*x^19 + 19209151138*x^18 - 20800159606*x^17 + 17768204859*x^16 - 12126221923*x^15 + 6718636422*x^14 - 3086566305*x^13 + 1204914514*x^12 - 407103232*x^11 + 118646908*x^10 - 28836372*x^9 + 5505383*x^8 - 758705*x^7 + 65305*x^6 - 2162*x^5 - 131*x^4 + 11*x^3)/(133600000*x^25 - 1875920000*x^24 + 12500686000*x^23 - 52604444000*x^22 + 156920670600*x^21 - 353103818000*x^20 + 622718972395*x^19 - 882777307660*x^18 + 1023713051333*x^17 - 983132187597*x^16 + 788634518440*x^15 - 531447118763*x^14 + 301890662895*x^13 - 144761728498*x^12 + 58568440406*x^11 - 19945788669*x^10 + 5692551701*x^9 - 1352405718*x^8 + 264899104*x^7 - 42210805*x^6 + 5370925*x^5 - 531418*x^4 + 39303*x^3 - 2039*x^2 + 66*x - 1).

A260727 Number of ways to place 3n rooks on an n X n board, with 3 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

Original entry on oeis.org

1, 23, 325, 3368, 28819, 218788, 1539399, 10314315, 66953292, 425761614, 2671506918, 16618186770, 102796975770, 633596982417, 3896224129259, 23924104985984, 146764696175937, 899809941054468, 5514653407814317, 33789681789605283, 207007665004469906

Offset: 2

Chris Cox, Jul 30 2015

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 3 balls when we can catch/throw up to 3 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1, inclusive.)

Colin Barker, Table of n, a(n) for n = 2..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Column k=3 of A269743.

Cf. A260575, A260582, A260583, A260584, A260585.

CoefficientList[Series[-(700*x^13 - 2435*x^12 + 4558*x^11 - 7532*x^10 + 10404*x^9 - 9697*x^8 + 5545*x^7 - 1844*x^6 + 336*x^5 - 39*x^4 + 7*x^3 - x^2)/((x^2)*(4000*x^14 - 35400*x^13 + 143100*x^12 - 349910*x^11 + 577675*x^10 - 680496*x^9 + 589248*x^8 - 380592*x^7 + 184037*x^6 - 66214*x^5 + 17423*x^4 - 3246*x^3 + 404*x^2 - 30*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jul 30 2015 *)

G.f.: -(700*x^13 - 2435*x^12 + 4558*x^11 - 7532*x^10 + 10404*x^9 - 9697*x^8 + 5545*x^7 - 1844*x^6 + 336*x^5 - 39*x^4 + 7*x^3 - x^2)/(4000*x^14 - 35400*x^13 + 143100*x^12 - 349910*x^11 + 577675*x^10 - 680496*x^9 + 589248*x^8 - 380592*x^7 + 184037*x^6 - 66214*x^5 + 17423*x^4 - 3246*x^3 + 404*x^2 - 30*x + 1).

A269742 Triangle of generalized Eulerian numbers T(n,k) = _2 read by rows, n >= 1, 0 <= k < 2*n.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 11, 4, 1, 1, 11, 72, 114, 72, 11, 1, 1, 26, 367, 1492, 2438, 1492, 367, 26, 1, 1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1, 1, 120, 6680, 109538, 727982, 2169674, 3107640, 2169674, 727982, 109538, 6680, 120, 1

Offset: 1

N. J. A. Sloane, Mar 16 2016

T(n,k) is the number of nonnegative integer n X n matrices with every row and column sum 2 and sum of entries below the main diagonal k. The case when every row and column sum is 1 is given by the Eulerian numbers (A008292). - Andrew Howroyd, Feb 22 2020

			Triangle begins:
  1;
  1, 1, 1;
  1, 4, 11, 4, 1;
  1, 11, 72, 114, 72, 11, 1;
  1, 26, 367, 1492, 2438, 1492, 367, 26, 1;
  1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1;
  ...
The matrices for row n=3, k=0..2 are:
  [2 0]  [1 1]  [0 2]
  [0 2]  [1 1]  [2 0]

Andrew Howroyd, Table of n, a(n) for n = 1..1600 (first 40 rows)
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Andrew Howroyd, PARI Program

Row sums are A000681.
Columns k=0..4 are A000012, A000295, A260585, A260575, A260582.
Central coefficients are A332729.
Cf. A008292, A269743, A269744.

\\ See link. - Andrew Howroyd, Feb 22 2020

Terms a(26) and beyond from Andrew Howroyd, Feb 22 2020

A260584 Number of ways to place 4n rooks on n X n board, 4 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

Original entry on oeis.org

1, 42, 1152, 22785, 358784, 4848569, 59160195, 674020718, 7332379979, 77311947872, 798116114567, 8122264310217, 81865063934240, 819786478839348, 8173571362926773, 81256681626746819, 806240597786756436, 7989356540290573170

Offset: 2

Jacob Landgraf, Jul 29 2015

nonn

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 4 balls when we can catch/throw up to 4 balls at a time. (Minimal in the sense that the throws are between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 2..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Cf. A260575, A260582, A260583, A260585, A260727.

CoefficientList[Series[-(25340000 x^23 -218339000 x^22 + 967516125 x^21 - 3006320955 x^20 + 7236534214 x^19 - 13729556248 x^18 + 20397063058 x^17 - 23597394968 x^16 + 21251854412 x^15 - 14962982713 x^14 + 8335966059 x^13 - 3793119227 x^12 + 1513380019 x^11 - 584800410 x^10 + 226357446 x^9 - 80585779 x^8 + 23590993 x^7 - 5268629 x^6 + 855872 x^5 - 97502 x^4 + 7700 x^3 - 464 x^2 + 26 x^1 - 1)/(196000000 x^26 - 2903600000 x^25 + 20460490000 x^24 - 91266464000 x^23 + 289327787000 x^22 - 693785336400 x^21 + 1307696973825 x^20 - 1987649503130 x^19 + 2479934403745 x^18 - 2572088215962 x^17 + 2237510543313 x^16 - 1642726164623 x^15 + 1021902480875 x^14 - 539757845397 x^13 + 242151721153 x^12 - 92151943921 x^11 + 29657096575 x^10 - 8031745172 x^9 + 1817290072 x^8 - 340120209 x^7 + 51938261 x^6 - 6350073 x^5 + 605172 x^4 - 43205 x^3 + 2168 x^2 -68 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 06 2015 *)

G.f.: -(25340000*x^25 - 218339000*x^24 + 967516125*x^23 - 3006320955*x^22 + 7236534214*x^21 - 13729556248*x^20 + 20397063058*x^19 - 23597394968*x^18 + 21251854412*x^17 - 14962982713*x^16 + 8335966059*x^15 - 3793119227*x^14 + 1513380019*x^13 - 584800410*x^12 + 226357446*x^11 - 80585779*x^10 + 23590993*x^9 - 5268629*x^8 + 855872*x^7 - 97502*x^6 + 7700*x^5 - 464*x^4 + 26*x^3 - x^2)/(196000000*x^26 - 2903600000*x^25 + 20460490000*x^24 - 91266464000*x^23 + 289327787000*x^22 - 693785336400*x^21 + 1307696973825*x^20 - 1987649503130*x^19 + 2479934403745*x^18 - 2572088215962*x^17 + 2237510543313*x^16 - 1642726164623*x^15 + 1021902480875*x^14 - 539757845397*x^13 + 242151721153*x^12 - 92151943921*x^11 + 29657096575*x^10 - 8031745172*x^9 + 1817290072*x^8 - 340120209*x^7 + 51938261*x^6 - 6350073*x^5 + 605172*x^4 - 43205*x^3 + 2168*x^2 - 68*x + 1).

cp's OEIS Frontend

A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.

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A260582 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

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A260583 Number of ways to place 3n rooks on an n X n board, 3 rooks in each row and each column, multiple rooks in an allowed cell, and exactly 4 rooks below the main diagonal.

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A260727 Number of ways to place 3n rooks on an n X n board, with 3 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

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A269742 Triangle of generalized Eulerian numbers T(n,k) = _2 read by rows, n >= 1, 0 <= k < 2*n.

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A260584 Number of ways to place 4n rooks on n X n board, 4 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

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