OFFSET
0,4
COMMENTS
A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.
The n-th row gives the coefficients of the matching-generating polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017
LINKS
Alois P. Heinz, Rows n = 0..13, flattened
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
EXAMPLE
T(2,1) = 6:
+---+ +---+ +---+ +---+ +---+ +---+
|o-o| | | |o | | o| |o | | o|
| | | | || | | || | \ | | / |
| | |o-o| |o | | o| | o| |o |
+---+ +---+ +---+ +---+ +---+ +---+
T(2,2) = 3:
+---+ +---+ +---+
|o-o| |o o| |o o|
| | || || | X |
|o-o| |o o| |o o|
+---+ +---+ +---+
Triangle T(n,k) begins:
1;
1;
1, 6, 3;
1, 20, 110, 180, 58;
1, 42, 657, 4890, 18343, 33792, 27380, 7416, 280;
1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, ...
...
MAPLE
b:= proc(n, l) option remember; local d, f, k;
d:= nops(l)/2; f:=false;
if n=0 then 1
elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])
else for k to d while not l[k] do od;
expand(b(n, subsop(k=f, l))+
`if`(k<d and n>1 and l[k+d+1],
x*b(n, subsop(k=f, k+d+1=f, l)), 0)+
`if`(k>1 and n>1 and l[k+d-1],
x*b(n, subsop(k=f, k+d-1=f, l)), 0)+
`if`(n>1 and l[k+d], x*b(n, subsop(k=f, k+d=f, l)), 0)+
`if`(k<d and l[k+1], x*b(n, subsop(k=f, k+1=f, l)), 0))
fi
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true$(n*2)])):
seq(T(n), n=0..7);
MATHEMATICA
b[n_, l_] := b[n, l] = Module[{d, f, k}, d = Length[l]/2; f = False; Which[ n == 0, 1, l[[1 ;; d]] == Table[f, d], b[n-1, Join[l[[d+1 ;; 2d]], Table[ True, d]]], True, For[k = 1, !l[[k]], k++]; Expand[b[n, ReplacePart[l, k -> f]] + If[k<d && n>1 && l[[k+d+1]], x*b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k>1 && n>1 && l[[k + d - 1]], x*b[n, ReplacePart[ l, {k -> f, k + d - 1 -> f}]], 0] + If[n>1 && l[[k + d]], x*b[n, ReplacePart[l, {k -> f, k+d -> f}]], 0] + If[k<d && l[[k+1]], x*b[n, ReplacePart[l, {k -> f, k+1 -> f}]], 0]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][
b[n, Table[True, 2n]]];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *)
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 04 2014
STATUS
approved