A187240 -id:A187240 - cp's OEIS frontend

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

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).

A189789 Number of ways to place 8 nonattacking bishops on an n x n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 147456, 3265920, 129024000, 1097712000, 12939264000, 66784798080, 436483031040, 1669619952000, 7629571031040, 23828156352000, 85476013572096, 230333593351680, 693478195200000, 1669577821632000

Offset: 1

Vaclav Kotesovec, Apr 27 2011

nonn

Bruno Berselli, Table of n, a(n) for n = 1..5000
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 (2, 14, -30, -90, 210, 350, -910, -910, 2730, 1638, -6006, -2002, 10010, 1430, -12870, 0, 12870, -1430, -10010, 2002, 6006, -1638, -2730, 910, 910, -350, -210, 90, 30, -14, -2, 1).

Cf. A177755, A177756, A177757, A177758, A177759, A178140, A187240

(* Number of ways to place k nonattacking bishops on an n x n toroidal board *)
tbishops[k_,n_]:=If[EvenQ[n],2^k*k!*Sum[Binomial[n/2,i]^2*Binomial[n/2,k-i]^2/Binomial[k,i],{i,0,k}],k!*Binomial[n,k]^2];
Table[tbishops[8,n],{n,1,20}] (* using k=8 for this sequence *)

a(n) = (1/80640) * (n-6)^2 * (n-4)^2 * (n-2)^2 * n^2 * (2*n^8 - 64*n^7 + 884*n^6 - 7048*n^5 + 37382*n^4 - 147904*n^3 + 468540*n^2 - 1108800*n + 1422225 + (28*n^6 - 840*n^5 + 10906*n^4 - 80640*n^3 + 370468*n^2 - 1034880*n + 1400175) * (-1)^n)

G.f.: 1152x^8*(35x^23 + 21178x^22 + 27889x^21 + 2133348x^20 + 3081175x^19 + 51948910x^18 + 72476645x^17 + 469213640x^16 + 538879520x^15 + 1803221880x^14 + 1580004720x^13 + 3146148264x^12 + 2014875632x^11 + 2544618104x^10 + 1144092320x^9 + 933224520x^8 + 278242005x^7 + 143723790x^6 + 25756935x^5 + 7854820x^4 + 693025x^3 + 104538x^2 + 2579x + 128) / ((1-x)^17*(x+1)^15)

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

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A189789 Number of ways to place 8 nonattacking bishops on an n x n toroidal 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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