A274105 Triangle read by rows: T(n,k) = number of configurations of k non-attacking bishops on the black squares of an n X n chessboard (0 <= k < n for n > 1).

1, 1, 1, 2, 1, 5, 4, 1, 8, 14, 4, 1, 13, 46, 46, 8, 1, 18, 98, 184, 100, 8, 1, 25, 206, 674, 836, 308, 16, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 41, 612, 4196, 13756, 20476, 11896, 1912, 32, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 61, 1440, 16940, 106772, 361780, 629336, 506600, 154256, 11600, 64

Offset

1,4

Comments

Rows give the coefficients of the independence polynomial of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

Links

Table of n, a(n) for n=1..67.

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]

Eric Weisstein's World of Mathematics, Black Bishop Graph

Eric Weisstein's World of Mathematics, Independence Polynomial

Example

Triangle begins:
1, 1,
1, 2,
1, 5, 4,
1, 8, 14, 4,
1, 13, 46, 46, 8,
1, 18, 98, 184, 100, 8,
1, 25, 206, 674, 836, 308, 16,
1, 32, 356, 1704, 3532, 2816, 632, 16,
1, 41, 612, 4196, 13756, 20476, 11896, 1912, 32,
1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32,
1, 61, 1440, 16940, 106772, 361780, 629336, 506600, 154256, 11600, 64,
...
Corresponding independence polynomials:
1+x, (K_1)
1+2*x, (P_2 = K_2)
1+5*x+4*x^2, (butterfly graph)
1+8*x+14*x^2+4*x^3,
...

Maple

with(combinat);
T:=n->add(stirling2(n+1, n+1-k)*x^k, k=0..n);
# bishops on black 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+1, 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

CoefficientList[Table[Sum[x^n Binomial[Ceiling[n/2], k] BellB[n - k, 1/x], {k, 0, Ceiling[n/2]}], {n, 10}], x] (* Eric W. Weisstein, Jun 26 2017 *)

Crossrefs

Alternate rows give A088960.

Row sums are A216332(n+1) for n>1.

Cf. A274106 (white squares), A288183, A201862, A002465.

Sequence in context: A345454 A271684 A194682 * A366156 A056242 A343960

Adjacent sequences: A274102 A274103 A274104 * A274106 A274107 A274108

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jun 14 2016

Status

approved