A002465 -id:A002465 - cp's OEIS frontend

A238261 Decimal expansion of a constant related to A187235.

4, 9, 1, 0, 8, 1, 4, 9, 6, 4, 5, 6, 8, 2, 5, 5, 8, 9, 8, 7, 5, 1, 5, 3, 4, 8, 0, 5, 2, 4, 0, 3, 5, 2, 1, 9, 7, 8, 9, 8, 7, 0, 5, 2, 8, 1, 7, 6, 7, 8, 4, 7, 1, 7, 6, 1, 3, 9, 4, 1, 1, 2, 0, 2, 2, 5, 6, 4, 1, 7, 8, 7, 7, 8, 7, 9, 9, 4, 7, 9, 7, 2, 9, 5, 1, 8, 1, 9, 7, 4, 1, 5, 3, 5, 5, 4, 4, 6, 1, 4, 2, 5, 0, 5, 3

Offset: 1

Vaclav Kotesovec, Feb 21 2014

			4.9108149645682558987515348...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.261.

Cf. A187235, A238262, A002465, A129256, A342111, A330287.

RealDigits[N[-(2*LambertW[-1,-1/2/Sqrt[E]])^2/(1+2*LambertW[-1,-1/2/Sqrt[E]]), 105]][[1]]

Equals lim n->infinity (A187235(n)/(n-1)!)^(1/n).

Equals -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)).

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).

A199033 Number of ways to place n non-attacking bishops on a 2 X 2n board.

Original entry on oeis.org

1, 4, 22, 128, 771, 4744, 29618, 186880, 1188679, 7608764, 48953224, 316283264, 2050706932, 13336273528, 86953633242, 568221290496, 3720529001823, 24403423540348, 160314652983158, 1054635453261568, 6946703172803003, 45809043607167328, 302395650703501688

Offset: 0

Vaclav Kotesovec, Nov 02 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
V. Kotesovec, Non-attacking chess pieces

Cf. A002465, A191236, A219197, A183160.

[(&+[Binomial(2*n-j+1,j)*Binomial(n+j+1,n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 19 2019

Table[Sum[Binomial[2n-j+1,j]*Binomial[n+j+1,n-j],{j,0,n}],{n,0,25}]

A199033(n):=sum(binomial(n+k+1, n-k)*binomial(2*n-k+1,k),k,0,n)$ makelist(A199033(n),n,0,22); /* Martin Ettl, Nov 15 2012 */

{a(n)=sum(k=0, n, binomial(n+k+1, n-k)*binomial(2*n-k+1, k))}

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(G^2/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 14 2012
for(n=0,25,print1(a(n),", "))

[sum(binomial(2*n-j+1,j)*binomial(n+j+1,n-j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 19 2019

Recurrence: (112*n^4 + 968*n^3 + 3048*n^2 + 4136*n + 2040)*a(n+2) = (728*n^4 + 5914*n^3 + 17550*n^2 + 22510*n + 10530)*a(n+1) + (189*n^4 + 1539*n^3 + 4578*n^2 + 5886*n + 2760)*a(n). - Vaclav Kotesovec, Oct 30 2011
a(n) = Sum_{j=0..n} (binomial(2n-j+1,j)*binomial(n+j+1,n-j)).
a(n) ~ 3^(3n+4)/2^(2n+5)/sqrt(3*Pi*n).
Self-convolution of A219197. - Paul D. Hanna, Nov 14 2012
G.f.: A(x) = G(x)^2 / (1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 14 2012
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n+2)). - Ilya Gutkovskiy, Oct 25 2017

Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 14 2012

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).

A189790 Number of ways to place n nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

1, 4, 6, 64, 120, 2304, 5040, 147456, 362880, 14745600, 39916800, 2123366400, 6227020800, 416179814400, 1307674368000, 106542032486400, 355687428096000, 34519618525593600, 121645100408832000, 13807847410237440000

Offset: 1

Vaclav Kotesovec, Apr 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..200
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf. A002465, A002454.

Table[If[EvenQ[n],2^n*((n/2)!)^2,n!],{n,1,20}]
Table[n!*SeriesCoefficient[1/(1-x)+x*ArcSin[x]/(1-x^2)^(3/2), {x,0,n}], {n,1,25}] (* Vaclav Kotesovec, Sep 26 2012 *)

a(n) = 2^n*((n/2)!)^2 if n is even and a(n) = n! if n is odd.
a(n) = n*(2*n-3)*a(n-2)-(n-3)*n*(n-2)^2*a(n-4). [Vaclav Kotesovec, Sep 26 2012]
E.g.f.: 1/(1-x)+x*arcsin(x)/(1-x^2)^(3/2). [Vaclav Kotesovec, Sep 26 2012]

A189791 Number of ways to place n nonattacking bishops on an 2n x 2n toroidal board.

Original entry on oeis.org

4, 80, 2688, 132864, 8647680, 699678720, 67711795200, 7629571031040, 981168437329920, 141817953779712000, 22760391875493888000, 4016046336733347840000, 772743693378451931136000, 161027573368536472485888000, 36127883615009765477842944000

Offset: 1

Vaclav Kotesovec, Apr 27 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf. A189790, A002465, A000172.

Table[2^n*n!*Sum[Binomial[n,i]^3,{i,0,n}],{n,1,20}]

a(n)=2^n*n!*Sum[Binomial[n,i]^3,{i,0,n}].
Asymptotic: a(n) ~ 2^(4n+1)*(n-1)!/Pi/sqrt(3) ~ 2^(4n+1)*n^n/exp(n)*sqrt(2/(3*Pi*n)).
Recurrence: a(n) = ((14*n^2-14*n+4)*a(n-1) + 32*(n-1)^3*a(n-2))/n.

A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 1, 8, 14, 4, 1, 12, 38, 32, 4, 1, 18, 98, 184, 100, 8, 1, 24, 188, 576, 652, 208, 8, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 40, 580, 3840, 12052, 16944, 9080, 1280, 16, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32

Offset: 0

N. J. A. Sloane, Jun 14 2016

From Eder G. Santos, Dec 16 2024: (Start)
The sequence counts every possible nonattacking configuration of k bishops on the white squares of an n X n chess board.
It is assumed that the n X n chess board has a black square in the upper left corner.
(End)

			Triangle begins:
  1;
  1;
  1,  2;
  1,  4,    2;
  1,  8,   14,     4;
  1, 12,   38,    32,     4;
  1, 18,   98,   184,   100,      8;
  1, 24,  188,   576,   652,    208,      8;
  1, 32,  356,  1704,  3532,   2816,    632,     16;
  1, 40,  580,  3840, 12052,  16944,   9080,   1280,     16;
  1, 50,  940,  8480, 38932,  89256,  93800,  37600,   3856,   32;
  1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32;
  ...
From _Eder G. Santos_, Dec 16 2024: (Start)
For example, for n = 3, k = 2, the T(3,2) = 2 nonattacking configurations are:
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
  |   |   |   | , | B |   | B |
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
(End)

Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
J. Perott, Sur le problème des fous, Bulletin de la S. M. F., tome 11 (1883), pp. 173-186.
Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas. arXiv:2411.16492 [math.CO], 2024. (considered as black board).
Eric Weisstein's World of Mathematics, White Bishop Graph.

Columns k=0-1 give: A000012, A007590.
Alternate rows give A088960.
Row sums are A216078(n+1).
T(2n,n) gives A191236.
T(2n+1,n) gives A217900(n+1).
T(n+1,n) gives A060546.
Cf. A274105 (black squares), A288182, A201862, A002465.

with(combinat): with(gfun):
T := n -> add(stirling2(n+1,n+1-k)*x^k, k=0..n):
# bishops on white squares
bish := proc(n) local m,k,i,j,t1,t2; global T;
    if n=0 then return [1] fi;
    if (n mod 2) = 0 then m:=n/2;
        t1:=add(binomial(m,k)*T(2*m-1-k)*x^k, k=0..m);
    else
        m:=(n-1)/2;
        t1:=add(binomial(m,k)*T(2*m-k)*x^k, k=0..m+1);
    fi;
    seriestolist(series(t1,x,2*n+1));
end:
for n from 0 to 12 do lprint(bish(n)); od:

T[n_] := Sum[StirlingS2[n+1, n+1-k]*x^k, {k, 0, n}];
bish[n_] := Module[{m, t1, t2}, If[Mod[n, 2] == 0,
   m = n/2;     t1 = Sum[Binomial[m, k]*T[2*m-1-k]*x^k, {k, 0, m}],
   m = (n-1)/2; t1 = Sum[Binomial[m, k]*T[2*m - k]*x^k, {k, 0, m+1}]];
CoefficientList[t1 + O[x]^(2*n+1), x]];
Table[bish[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 25 2022, after Maple code *)

def stirling2_negativek(n, k):
  if k < 0: return 0
  else: return stirling_number2(n, k)
print([sum([binomial(floor(n/2), j)*stirling2_negativek(n-j, n-k) for j in [0..k]]) for n in [0..10] for k in [0..n-1+kronecker_delta(n,0)]]) # Eder G. Santos, Dec 01 2024

From Eder G. Santos, Dec 01 2024: (Start)
T(n,k) = Sum_{j=0..k} binomial(floor(n/2),j) * Stirling2(n-j,n-k).
T(n,k) = T(n-1,k) + (n-k+1-A000035(n)) * T(n-1,k-1), T(n,0) = 1, T(0,k) = delta(k,0). (End)

T(0,0) prepended by Eder G. Santos, Dec 01 2024

A238262 Decimal expansion of a multiplicative constant related to A187235.

Original entry on oeis.org

2, 4, 2, 5, 2, 1, 9, 1, 2, 8, 1, 5, 2, 3, 5, 9, 8, 5, 9, 4, 9, 3, 2, 1, 0, 8, 0, 3, 8, 6, 3, 9, 2, 0, 2, 9, 5, 1, 3, 8, 3, 2, 8, 7, 2, 3, 5, 3, 2, 7, 6, 1, 2, 1, 1, 5, 4, 1, 0, 1, 7, 8, 0, 6, 6, 8, 7, 0, 5, 1, 9, 4, 8, 3, 8, 5, 5, 0, 9, 5, 1, 1, 5, 9, 2, 0, 4, 4, 5, 3, 9, 3, 9, 9, 8, 0, 6, 5, 5, 0, 4, 2, 0, 4

Offset: 0

Vaclav Kotesovec, Feb 21 2014

			0.242521912815235985949321...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.261.

Cf. A187235, A238261, A002465.

Equals lim n->infinity A187235(n) / ((n-1)! * A238261^n).

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).

cp's OEIS Frontend

A238261 Decimal expansion of a constant related to A187235.

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

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A199033 Number of ways to place n non-attacking bishops on a 2 X 2n 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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A189790 Number of ways to place n nonattacking bishops on an n X n toroidal board.

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A189791 Number of ways to place n nonattacking bishops on an 2n x 2n toroidal board.

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A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

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