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A292080 Number of nonequivalent ways to place n non-attacking rooks on an n X n board with no rook on 2 main diagonals up to rotations and reflections of the board. 2
1, 0, 0, 0, 2, 2, 14, 84, 630, 6096, 55336, 672160, 7409300, 104999520, 1366363752, 22068387264, 331233939624, 6005919062528, 102144359744192, 2054811316442112, 39053339674065360, 863259240785840640, 18132529836143846560, 436899062862222484480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

For odd n, there are no symmetrical configurations of non-attacking rooks without a rook in the main diagonal, so a(2n+1) = A003471(2n+1) / 8. For even n, configurations with rotational and diagonal symmetry are possible.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

Andrew Howroyd, Nonequivalent and Symmetric Solutions

FORMULA

a(2n+1) = A003471(2n+1) / 8, a(2n) = (A003471(2n) + 2^n * A000166(n) + 2*A037224(2*n) + 2*A053871(n)) / 8.

EXAMPLE

Case n=4: The 2 nonequivalent solutions are:

   _ x _ _     _ x _ _

   x _ _ _     _ _ _ x

   _ _ _ x     x _ _ _

   _ _ x _     _ _ x _

Case n=5: The 2 nonequivalent solutions are:

   _ x _ _ _   _ x _ _ _

   x _ _ _ _   _ _ _ _ x

   _ _ _ x _   x _ _ _ _

   _ _ _ _ x   _ _ x _ _

   _ _ x _ _   _ _ _ x _

MATHEMATICA

sf[n_] := n! * SeriesCoefficient[Exp[-x ] / (1 - x), {x, 0, n}];

F[n_] := (Clear[v]; v[_] = 0; For[m = 4, m <= n, m++, v[m] = (m - 1)*v[m - 1] + 2*If[OddQ[m], (m - 1)*v[m - 2], (m - 2)*If[m == 4, 1, v[m - 4]]]]; v[n]);

d[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(2k)!/(2^k*k!), {k, 0, n}];

R[n_] := If[OddQ[n], 0, (n - 1)!*2/(n/2 - 1)!];

a[0] = 1; a[n_] := (F[n] + If[OddQ[n], 0, m = n/2; 2^m * sf[m] + 2*R[m] + 2*d[m]])/8;

Table[a[n], {n, 0, 23}] (* Jean-Fran├žois Alcover, Dec 28 2017, after Andrew Howroyd *)

PROG

(PARI) \\ here sf is A000166, F is A003471, D is A053871, R(n) is A037224(2n).

sf(n) = {n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}

F(n) = {my(v = vector(n)); for(n=4, length(v), v[n]=(n-1)*v[n-1]+2*if(n%2==1, (n-1)*v[n-2], (n-2)*if(n==4, 1, v[n-4]))); v[n]}

D(n) = {sum(k=0, n, (-1)^(n-k) * binomial(n, k) * (2*k)!/(2^k*k!))}

R(n) = {if(n%2==1, 0, (n-1)!*2/(n/2-1)!)}

a(n) = {(F(n) + if(n%2==1, 0, my(m=n/2); 2^m * sf(m) + 2*R(m) + 2*D(m)))/8}

CROSSREFS

Cf. A000166, A000903, A003471, A037224, A053871, A064280.

Sequence in context: A321179 A166114 A202730 * A333372 A291376 A273319

Adjacent sequences:  A292077 A292078 A292079 * A292081 A292082 A292083

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Sep 12 2017

STATUS

approved

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Last modified September 28 21:51 EDT 2020. Contains 337414 sequences. (Running on oeis4.)