A243424 Triangle T(n,k) read by rows of number of ways k domicules can be placed on an n X n square (n >= 0, 0 <= k <= floor(n^2/2)).

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, 34431880, 16172160, 3219364, 170985, 1, 110, 5375, 154434, 2911226, 38049764, 355340561, 2408715568

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 *)

Crossrefs

Columns k=0-5 give: A000012, A002943(n-1) for n>0, A243464, A243465, A243466, A243467.

Row sums give A220638.

T(n,floor(n^2/2)) gives A243510.

T(n,floor(n^2/4)) gives A243511.

Cf. A242861 (the same for dominoes), A239264.

Sequence in context: A240264 A119743 A272643 * A182227 A108451 A122178

Adjacent sequences: A243421 A243422 A243423 * A243425 A243426 A243427

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 04 2014

Status

approved