A187239 -id:A187239 - cp's OEIS frontend

A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

1, 2, 9, 70, 729, 9918, 167281, 3423362, 82609921, 2319730026, 74500064809, 2711723081550, 110568316431609, 5016846683306758, 251180326892449969, 13806795579059621930, 827911558468860287041, 53940895144894708523922, 3799498445458163685753481, 288400498147873552894868886

Offset: 0

Vaclav Kotesovec, Dec 06 2011

nonn

Also the number of vertex covers and independent vertex sets of the n X n bishop graph.

Vaclav Kotesovec, Table of n, a(n) for n = 0..320
Vaclav Kotesovec, Non-attacking chess pieces
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A010790, A172123, A172124, A172127, A172129, A176886, A187239 - A187242, A002465, A216078, A216332.

Row sums of A378590.

knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!
*Sum[Binomial[2n-k,n-k+i]*Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,1,n-i}]
*Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,1,n+i-k}],{i,Max[0,k-n],Min[k,n]}]]);
Table[1+Sum[knbishops[k,n],{k,1,2n-1}],{n,1,25}]

a(n) = A216078(n+1) * A216332(n+1). - Andrew Howroyd, May 08 2017

a(0)=1 prepended by Alois P. Heinz, Dec 01 2024

A201248 Number of ways to place 7 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 0, 216, 38070, 1314600, 21191208, 207830308, 1442794332, 7775083960, 34530764200, 131660992164, 443702617356, 1350258600008, 3771242866680, 9789675562020, 23856321869260, 55015308882264, 120855465245464, 254284702668580, 514791197224860, 1006655249550696

Offset: 1

Vaclav Kotesovec, Nov 28 2011

nonn

Fers is a leaper [1,1].

V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Cf. A187239, A201243, A201244, A201245, A201246, A201247.

a(n) = n^14/5040 - n^12/48 + n^11/30 + 673n^10/720 - 17n^9/6 - 1019n^8/48 + 197n^7/2 + 9772n^6/45 - 3443n^5/2 + 47n^4/4 + 74259n^3/5 - 1816352n^2/105 - 49376n + 90660, n>=6.

G.f.: 2*x^4*(125*x^16 - 1785*x^15 + 11715*x^14 - 50121*x^13 + 158605*x^12 - 367485*x^11 + 570175*x^10 - 533381*x^9 + 460395*x^8 - 1262515*x^7 + 2731225*x^6 - 1795227*x^5 - 5484089*x^4 - 2685639*x^3 - 383115*x^2 - 17415*x - 108)/(x-1)^15.

A187240 Number of ways to place 8 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 32, 12944, 867328, 22522960, 328097824, 3209594096, 23460698496, 137045115696, 670158151296, 2835083100640, 10634260782464, 36033282628832, 111923478184128, 322412415716896, 869530617762304, 2212626780591008, 5346773160475488, 12336574243905648, 27303885052866048

Offset: 1

Vaclav Kotesovec, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
E. Weisstein, Bishops Problem, MathWorld
Index entries for linear recurrences with constant coefficients, signature (6, -4, -46, 95, 116, -496, 44, 1331, -990, -2068, 2838, 1683, -4488, 0, 4488, -1683, -2838, 2068, 990, -1331, -44, 496, -116, -95, 46, 4, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886, A187239.

CoefficientList[Series[- 16 x^4 (2520 x^22 + 47160 x^21 + 808884 x^20 + 7825113 x^19 + 54648810 x^18 + 265795497 x^17 + 965510650 x^16 + 2638742416 x^15 + 5598377728 x^14 + 9280070520 x^13 + 12189441400 x^12 + 12689244954 x^11 + 10499675700 x^10 + 6853251794 x^9 + 3501200340 x^8 + 1373620536 x^7 + 404231224 x^6 + 85610168 x^5 + 12313860 x^4 + 1085765 x^3 + 49362 x^2 + 797 x + 2) / ((x - 1)^17 (x + 1)^11), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^16/40320 - n^15/1080 + 7n^14/432 - 1153n^13/6480 + 53951n^12/38880 - 187277n^11/22680 + 106928053n^10/2721600 - 13957093n^9/90720 + 182160427n^8/362880 - 8821499n^7/6480 + 1176831457n^6/388800 - 490477369n^5/90720 + 8235592409n^4/1088640 - 726205757n^3/90720 + 1815275047n^2/302400 - 7953419n/2880 + 8491/16 + (-n^10/960 + 5n^9/144 - 307n^8/576 + 1793n^7/360 - 90571n^6/2880 + 201911n^5/1440 - 513865n^4/1152 + 477841n^3/480 - 4271471n^2/2880 + 1269721n/960 - 8491/16)*(-1)^n.
G.f.: -16x^5*(2520x^22 + 47160x^21 + 808884x^20 + 7825113x^19 + 54648810x^18 + 265795497x^17 + 965510650x^16 + 2638742416x^15 + 5598377728x^14 + 9280070520x^13 + 12189441400x^12 + 12689244954x^11 + 10499675700x^10 + 6853251794x^9 + 3501200340x^8 + 1373620536x^7 + 404231224x^6 + 85610168x^5 + 12313860x^4 + 1085765x^3 + 49362x^2 + 797x + 2)/((x-1)^17*(x+1)^11).
a(8) = A002465(8).

A187241 Number of ways to place 9 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 1600, 389312, 22057472, 565532992, 8611750848, 90564534336, 720227187456, 4603893554496, 24675964279680, 114402835995392, 469601097840640, 1737913582100864, 5882030372643968, 18417596366384512, 53854324059153920, 148209412582029184, 386390343290393024, 959556901097413696

Offset: 1

Vaclav Kotesovec, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
Index entries for linear recurrences with constant coefficients, signature (6, -2, -58, 102, 214, -690, -234, 2418, -962, -5226, 4862, 7150, -11154, -5434, 16302, 0, -16302, 5434, 11154, -7150, -4862, 5226, 962, -2418, 234, 690, -214, -102, 58, 2, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886, A187239, A187240.

CoefficientList[Series[- 64 x^5 (5670 x^25 + 116100 x^24 + 2282283 x^23 + 25883910 x^22 + 220244661 x^21 + 1330673229 x^20 + 6121839129 x^19 + 21511823232 x^18 + 59645434477 x^17 + 131494649245 x^16 + 234424379246 x^15 + 339339084372 x^14 + 401937236082 x^13 + 389328811002 x^12 + 308645316626 x^11 + 199052247464 x^10 + 103780570480 x^9 + 43151321222 x^8 + 14078209111 x^7 + 3508317590 x^6 + 644755881 x^5 + 82579449 x^4 + 6782181 x^3 + 308200 x^2 + 5933 x + 25) / ((x - 1)^19 (x + 1)^13), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^18/362880 - n^17/7560 + 181n^16/60480 - 14509n^15/340200 + 2101n^14/4860 - 101071n^13/30240 + 112406401n^12/5443200 - 143351879n^11/1360800 + 2465350549n^10/5443200 - 14081834n^9/8505 + 55888723201n^8/10886400 - 6055816813n^7/453600 + 155816526107n^6/5443200 - 13489156949n^5/272160 + 183801705823n^4/2721600 - 15816472541n^3/226800 + 30820237351n^2/604800 - 919392091n/40320 + 1101239/256 + (-n^12/5760 + 11n^11/1440 - 113n^10/720 + 51793n^9/25920 - 202873n^8/11520 + 3428791n^7/30240 - 1050169n^6/1920 + 8590259n^5/4320 - 1034689n^4/192 + 68481311n^3/6480 - 81534479n^2/5760 + 465686363n/40320 - 1101239/256)*(-1)^n.
G.f.: -64x^6*(5670x^25 + 116100x^24 + 2282283x^23 + 25883910x^22 + 220244661x^21 + 1330673229x^20 + 6121839129x^19 + 21511823232x^18 + 59645434477x^17 + 131494649245x^16 + 234424379246x^15 + 339339084372x^14 + 401937236082x^13 + 389328811002x^12 + 308645316626x^11 + 199052247464x^10 + 103780570480x^9 + 43151321222x^8 + 14078209111x^7 + 3508317590x^6 + 644755881x^5 + 82579449x^4 + 6782181x^3 + 308200x^2 + 5933x + 25)/((x-1)^19*(x+1)^13).
a(9) = A002465(9).

A187242 Number of ways to place 10 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 64, 81184, 12448832, 627961728, 15915225216, 251806066272, 2814607288320, 24088436720256, 166645918174848, 969258913391552, 4878776675787392, 21731689658569984, 87161301448676352, 319192073724720448, 1079363369445639936, 3401826465353378560, 10070308904424957632, 28183638842590122720

Offset: 1

Vaclav Kotesovec, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
Index entries for linear recurrences with constant coefficients, signature (6, 0, -70, 105, 336, -896, -720, 3900, -280, -10752, 6552, 20020, -21840, -24960, 43472, 18018, -60060, 0, 60060, -18018, -43472, 24960, 21840, -20020, -6552, 10752, 280, -3900, 720, 896, -336, -105, 70, 0, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886, A187239 - A187241.

CoefficientList[Series[- 32 x^5 (113400 x^29 + 2518560 x^28 + 55426428 x^27 + 713122128 x^26 + 7133734665 x^25 + 51575533686 x^24 + 289157705424 x^23 + 1253334719652 x^22 + 4339842816598 x^21 + 12089938835312 x^20 + 27595185140132 x^19 + 51899069651452 x^18 + 81237872407883 x^17 + 106097483667238 x^16 + 116126611566624 x^15 + 106417824457960 x^14 + 81632991696988 x^13 + 52161861060464 x^12 + 27621327391332 x^11 + 11998025297736 x^10 + 4224689442543 x^9 + 1183463783138 x^8 + 257650398544 x^7 + 42074808244 x^6 + 4911799606 x^5 + 379785344 x^4 + 17289788 x^3 + 373804 x^2 + 2525 x + 2) / ((x - 1)^21 (x + 1)^15), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^20/3628800 - n^19/60480 + 341n^18/725760 - 45949n^17/5443200 + 235433n^16/2177280 - 308291n^15/291600 + 14982871n^14/1814400 - 43484267n^13/816480 + 175706737n^12/604800 - 2444962049n^11/1796256 + 30003106793n^10/5443200 - 44899907477n^9/2332800 + 9919713547n^8/172800 - 18390588424n^7/127575 + 217346831209n^6/725760 - 8233418533709n^5/16329600 + 104224385179n^4/155520 - 14600765627n^3/21600 + 583132621007n^2/1209600 - 46669993739n/221760 + 19990663/512 + (-n^14/40320 + n^13/720 - 1267n^12/34560 + 15721n^11/25920 - 730663n^10/103680 + 5532407n^9/90720 - 98193341n^8/241920 + 10640393n^7/5040 - 99209431n^6/11520 + 1417368727n^5/51840 - 686809973n^4/10368 + 2144839679n^3/18144 - 1683044471n^2/11520 + 2242597633n/20160 - 19990663/512)*(-1)^n.
G.f.: -32x^6*(113400x^29 + 2518560x^28 + 55426428x^27 + 713122128x^26 + 7133734665x^25 + 51575533686x^24 + 289157705424x^23 + 1253334719652x^22 + 4339842816598x^21 + 12089938835312x^20 + 27595185140132x^19 + 51899069651452x^18 + 81237872407883x^17 + 106097483667238x^16 + 116126611566624x^15 + 106417824457960x^14 + 81632991696988x^13 + 52161861060464x^12 + 27621327391332x^11 + 11998025297736x^10 + 4224689442543x^9 + 1183463783138x^8 + 257650398544x^7 + 42074808244x^6 + 4911799606x^5 + 379785344x^4 + 17289788x^3 + 373804x^2 + 2525x + 2)/((x-1)^21*(x+1)^15).
a(10) = A002465(10).

A378590 Total number of ways to place k nonattacking bishops on an n X n chess board. Triangle T(n,k) read by rows (0 <= k <= 2*n-[n>0]-[n>1]).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 26, 26, 8, 1, 16, 92, 232, 260, 112, 16, 1, 25, 240, 1124, 2728, 3368, 1960, 440, 32, 1, 36, 520, 3896, 16428, 39680, 53744, 38368, 12944, 1600, 64, 1, 49, 994, 10894, 70792, 282248, 692320, 1022320, 867328, 389312, 81184, 5792, 128

Offset: 0

Eder G. Santos, Dec 01 2024

The sequence counts every possible nonattacking configuration of k bishops on an n x n chess board.

			Triangle begins:
  1;
  1  1;
  1  4   4;
  1  9  26    26     8;
  1 16  92   232   260    112     16;
  1 25 240  1124  2728   3368   1960     440     32;
  1 36 520  3896 16428  39680  53744   38368  12944   1600    64;
  1 49 994 10894 70792 282248 692320 1022320 867328 389312 81184 5792 128;
  ...
For example, for n = 2, k=2, the T(2,2)=4 nonattacking configurations are:
  +---+---+   +---+---+   +---+---+   +---+---+
  | B | B |   | B |   |   |   | B |   |   |   |
  +---+---+ , +---+---+ , +---+---+ , +---+---+
  |   |   |   | B |   |   |   | B |   | B | B |
  +---+---+   +---+---+   +---+---+   +---+---+

S. Chaiken, C. R. H. Hanusa, and T. Zaslavsky, A q-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders, J. Korean Math. Soc., 57(6): 1407-1433, 2020; see also arXiv preprint, arXiv:1609.00853 [math.CO], 2016-2020.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 234-259.
Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024.

Columns k=0-1 give: A000012, A000290.
Columns k=2-10 for n>=1 give: A172123, A172124, A172127, A172129, A176886, A187239, A187240, A187241, A187242.
Main diagonal T(n,n) gives A002465.
Row sums give A201862.
Cf. A000079.

def stirling2_negativek(n,k):
  if k < 0: return 0
  else: return stirling_number2(n,k)
print([sum([sum([binomial(floor(n/2),i)*stirling2_negativek(n-i,n-j)*sum([binomial(ceil(n/2),l)*stirling2_negativek(n-l,n-k+j) for l in [0..k-j]]) for i in [0..j]]) for j in [0..k]]) for n in [0..10] for k in [0..2*n-2+kronecker_delta(n,1)+2*kronecker_delta(n,0)]])

T(n,k) = Sum_{j=0..k} (Sum_{i=0..j} binomial(floor(n/2),i) * Stirling2(n-i,n-j)) * (Sum_{l=0..k-j} binomial(ceiling(n/2),l) * Stirling2(n-l,n-k+j)).

T(n,2*n-2+delta(n,1)+2*delta(n,0)) = A000079(n)-delta(n,1).

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A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

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A201248 Number of ways to place 7 non-attacking ferses on an n X n board.

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A187240 Number of ways to place 8 nonattacking bishops on an n X n board.

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A187241 Number of ways to place 9 nonattacking bishops on an n X n board.

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A187242 Number of ways to place 10 nonattacking bishops on an n X n board.

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A378590 Total number of ways to place k nonattacking bishops on an n X n chess board. Triangle T(n,k) read by rows (0 <= k <= 2*n-[n>0]-[n>1]).

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