A274106 Triangle read by rows: T(n,k) = number of configurations of k non-attacking bishops on the white squares of an n X n chessboard (0 <= k < n).

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

1,3

Links

Table of n, a(n) for n=1..66.

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.

Eric Weisstein's World of Mathematics, White Bishop Graph.

Example

Triangle begins:
  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;
  ...

Maple

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 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 1 to 12 do lprint(bish(n)); od:

Mathematica

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 *)

Crossrefs

Alternate rows give A088960.

Row sums are A216078(n+1).

Cf. A274105 (black squares), A288182, A201862, A002465.

Sequence in context: A123486 A350161 A158264 * A354802 A158982 A127124

Adjacent sequences: A274103 A274104 A274105 * A274107 A274108 A274109

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 14 2016

Status

approved