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
Examples
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, ... ...
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
Crossrefs
Programs
-
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`(k1 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 (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[k1 && 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 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 *)
Comments